Numerical Solution of Partial Differential Equations

Transcription

1 Numerical Solution of Partial Differential Equations Prof. Ralf Hiptmair, Prof. Christoph Schwab und Dr. H. Harbrecht V1.0: summer term 2004, V2.0: winter term 2005/2006 Draft version October 26, 2005 (C) Seminar für Angewandte Mathematik, ETH Zürich 0.0 p. 1

2 Contents 1 Second-order scalar elliptic boundary value problems Classification of boundary value problems Stationary heat conduction Boundary conditions Characteristics of elliptic boundary value problems Weak derivatives Variational formulation of boundary value problem Functional framework Essential and natural boundary conditions The Dirichlet principle p. 2

3 2 The Finite Element Method (FEM) Fundamentals Galerkin discretization The (linear) algebraic setting Principles of FEM Linear H 1 -conforming finite elements Simplicial Lagrangian finite elements Parametric finite elements Lagrangian finite elements on quadrilaterals/hexahedra Degrees of freedom Finite Element Implementation Mesh file format Assembly Mesh data structures Algorithms Local computations Numerical quadrature Treatment of essential boundary conditions Boundary approximation Treatment of hanging nodes Static condensation p. 3

4 2.3 Finite Difference Methods (FDM) From FEM to FD The discrete maximum principle Finite difference convergence theory Finite Volume Methods (FVM) Principles of FVM Dual meshes From FVM to FEM Finite Element Convergence Theory A priori error estimates The Sobolev scales The Bramble-Hilbert lemma Transformation techniques Interpolation error estimates Elliptic regularity theory Convergence of finite element solutions Variational crimes Abstract theory Impact of numerical quadrature Approximation of boundary Duality estimates Pointwise estimates p. 4

5 Index 241 Keywords Examples Definitions MATLAB-CODE Symbols Proof of Lemma Reporting errors Please report any error or dubious manipulation/assertion/reasoning by ! Examples: From: "MrX" To: Subject: NPDE05: Error Error on page XX, Section XX, Formula (XX): 0.0 index i has to be changed to j p. 5

6 From: "MrX" To: Subject: NPDE05: Error Page XX, Section XX, Theorem XX: the sign in front of the \Psi seems to be wrong Preamble This course is part of the Computational Science and Engineering (CSE) curriculum. Main skills to be acquired in this course: Ability to implement advanced numerical methods for the solution of partial differential equations in MATLAB efficiently Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations 0.0 p. 6

7 Ability to select and assess numerical methods in light of the predictions of theory Ability to identify features of a model that are relevant for the selection and performance of a numerical algorithm Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations. This course Numerical analysis of PDE ( mathematics curriculum) Instruction on how to apply software packages 0.0 p. 7

8 1 Second-order scalar elliptic boundary value problems 1.1 Classification of boundary value problems Boundary value problem Given a partial differential operator L, a domain R d, a boundary differential operator B, boundary values g, and a source term f, seek a function u : R q such that L(u) = f in, B(u) =g on part of boundary. Three main categories of boundary value problems (BVPs) for partial differential equations (PDE): p

9 Elliptic BVPs Parabolic initial boundary value problems (IBVPs) Hyperbolic IBVPs, among them wave propagation problems and conservation laws. Rigorous mathematical definition Physics behind PDE-based models Further discussion of classification: [2, 1] and [5, Ch. 1] What are second-order scalar elliptic boundary value problems? Name: second-order : PDE features second spatial derivatives scalar : Unknown is a function u : R elliptic : equilibrium character (see following sections) 1.2 p. 9

10 1.2 Stationary heat conduction R 3 : bounded open region occupied by solid object: computational domain Fourier s law j(x) = σ (x) grad u(x), x. (1.2.1) Meaning of quantities: j = heat flux ([j] = 1 W m 2) u = temperature ([u] = 1K) σ = heat conductivity ([σ ] = 1 W Km ) (1.2.1) Heat flow from hot to cold regions linearly proportional to gradient of temperature σ : σ = σ (x) for non-homogeneous materials (spatially varying heat conductivity) σ can even be discontinuous for composite materials 1.2 p. 10

11 From thermodynamic principles: σ,σ + > 0: 0 < σ σ (x) σ + < for almost all x. (1.2.2) Terminology: (1.2.2) σ = uniformly positive Remark In the case of Anisotropic materials: σ (x) R 3,3, σ (x) = σ (x) T, x heat conductivity tensor Example (Anisotropic material). Metal wires in polymer matrix modelled by effective heat conductivity: (1.2.2) becomes: σ,σ + > 0: σ ξ 2 ξ T σ (x)ξ σ + ξ 2 ξ R 3, for almost all x PSfrag (1.2.3). replacements Terminology: σ is uniformly symmetric positive definite Def j grad u 1.2 p. 11

12 V Conservation of energy j n ds = f dx for all control volumes V (1.2.4) V f = heat source/sink ([ f ] = W m 3), f = f (x) and f can be discontinuous By Gauss theorem we get local form of energy conservation: div j = f. (1.2.5) } {{ } Combine equations j = σ (x) grad u (1.2.1) + div j = f (1.2.5) div(σ grad u) = f in (1.2.6) Linear scalar second order elliptic PDE 1.2 p. 12

13 Remark (Scaling). Before numerical treatment: conversion into non-dimensional form by scaling Recaling of length : x = x/l, L = characteristic length temperature : û = u/t, T = characteristic temperature difference heat source : f = f/f, F = typical heat source ( ) T σ div x L 2 F grad x û = f in. (one non-dimensional characteristic parameter remains) Note: Notations retained for scaled quantities Remark If σ const., by rescaling of (1.2.6): u = f in. (1.2.7) = div grad = 2 x x x 2 3 = Laplace operator (1.2.7) is called Poisson equation 1.3 p. 13

14 1.3 Boundary conditions Boundary conditions on surface/boundary of : (i) Temperature u is fixed: with g : R prescribed u = g on. (1.3.1) Dirichlet boundary conditions (ii) Heat flux j through is fixed: vectorfield on with h : R prescribed, n : R 3 exterior unit normal j n = h on. (1.3.2) Neumann boundary conditions (iii) Heat flux through depends on (local) temperature: with increasing function : R R j n = (u) on (1.3.3) radiation boundary conditions Example (Convective cooling (simple model)). j n = q(u u 0 ) on, where 0 < q q(x) q + < for almost all x. 1.3 p. 14

15 Example (Radiative cooling (simple model)). j n = α u u 0 (u u 0 ) 3 on, with α > 0 Non-linear boundary condition Terminology: If g = 0 or h = 0 homogeneous Dirichlet or Neumann boundary conditions Remark (mixed boundary conditions). Different boundary conditions can be prescribed on different parts of ( mixed boundary conditions) Ɣ R Ɣ N Ɣ D Example ( Wrapped rock on a stove ). Non-homogeneous Dirichlet boundary conditions on Ɣ D Homogeneous Neumann boundary condtions on Ɣ N Convective cooling boundary conditions on Ɣ R Partition: = Ɣ D Ɣ N Ɣ R, Ɣ D, Ɣ N, Ɣ R mutually disjoint }{{} div(σ grad u) = f + boundary conditions elliptic boundary value problem (BVP) For second order elliptic boundary value problems exactly one boundary condition is needed on any part of. 1.3 p. 15

16 Remark Solution operator ( ) f g u for (1.2.6), (1.3.1) is linear 1.4 Characteristics of elliptic boundary value problems Qualitative insights gained from heat conduction model: continuity: the temperature u must be continuous (jump in u j = ). normal component of j across surfaces inside must be continuous (jump in j n heat source f of infinite intensity). interior smoothness of u: u smooth where f and σ smooth. non-locality: local alterations in f, g, h affect u everywhere in. quasi-locality: If local changes in f, g, h confined to, their effects decay away from. p

17 maximum principle: (in the absence of heat sources extremal temperatures on the boundary) if f 0, then inf u(y) u(x) sup u(y) y y for all x }{{} Typical features of solutions of elliptic boundary value problems Example (Scalar elliptic boundary value problem in one space dimension). Poisson equation (1.2.7) in 1D: u = f f discontinuous, piecewise C 0 u C 1, piecewise C 2 Example (Smoothness of solution of scalar elliptic boundary value problem). u = f (x) in :=]0, 1[ 2, u = 0 on, (1.4.1) f (x) := sign(sin(2πk 1 x 1 ) sin(2πk 2 x 2 )), x, k 1, k 2 N. Approximate solution computed by means of linear Lagrangian finite elements + lumping ( Sect. 2, details in Sect , 2.2.6) 1.4 p. 17

18 1 0.5 f y Source term f (x), k 1 = k 2 = 2 x Fig. 1 Solution of (1.4.1) Fig. 2 Smooth u despite rough f! Example (Quasi-locality of solution of scalar elliptic boundary value problem). 1.4 p. 18

19 u = f δ (x) in :=]0, 1[ 2, u = 0 on, (1.4.2) δ 2, if x ( 1/2 f δ (x) = 1/2) 2 δ,, δ > 0. (1.4.3) 0 elsewhere Cross section of solution u δ = δ = δ = δ = δ = u(x,0.5) Fig x Fig p. 19

20 1.5 Weak derivatives Assume translation symmetry in two coordinate directions ( x 2 = x 3 = 0) 1D stationary heat conduction model ( Ex ): d d (σ (x) dx dx u) = 0, u(0) = 1, u(1) = u 1 R, { σ where σ (x) = 1 for 0 < x < 2 1, σ 2 for 1 2 < x < 1. u(x) = (Heat conduction in a big flat wall) PSfrag replacements u 2u 1 σ 2 x for 0 < x < 1 σ 1 + σ 2, σ (x) = σ 1 2 2u 1 σ 1 (x 1) + u 1 for 1 σ 1 + σ 2 < x < 1. 2 σ (x) = σ 2 1 Note: For discontinuous σ (normal) continuity of j rules out continuous differentiability of u x How to make sense of 1.5 d dx u? p. 20

21 Answer: weak derivative (piecewise derivative almost everywhere) Definition (Weak derivative). The weak partial derivative u x j, j = 1,..., d (derivative in the sense of distributions), of a locally integrable function u : R d R is, if it exists, a locally integrable function j u : R that satisfies j u v dx = u v dx v C x 0 ( ), j where C 0 ( ) is the space of compactly supported smooth functions R. [Multidimensional integration by parts formula] Weak derivative is genuine generalization of classical derivative Example (Weak derivative in 1D). u(x) = { u 1 (x) for 0 < x < 1 2, u 2 (x) for 1 2 x < 1, with u 1, u 2 smooth, u 1 (1/2) = u 2 (1/2). 1.5 p. 21

22 We show that in this case that is weak derivative = piecewise derivative { du dx = u 1 (x) for 0 < x < 1 2, u 2 (x) for 1 2 x < 1. for v C du dx v dx = 1 /2 0 u 1 v dx + 1 = [u 1 v] 1 /2 0 1 /2 (]0, 1[) v(0) = v(1) = /2 u 2 v dx = (u } 1 (1/2) u {{ 2 (1/2))v(1/2) } =0 u 1 v dx + [u 2 v] 1 1/ uv dx, 1/2 u 2 v dx fits definition of weak derivative Remark Weak x j weak differential operators grad u := ( x u,..., u 1 x ) T, d div j := j 1 x + + j d 1 x, d u, curl j,... Below: All differential operators will be understood in weak sense 1.5 p. 22

23 New interpretation of a partial differential equation PDE in weak sense PDE in classical sense 1.6 Variational formulation of boundary value problem Formal approach: STEP 1: test PDE with smooth functions (cf. weak derivative, Def ) STEP 2: integrate over domain (cf. weak derivative, Def ) STEP 3: perform integration by parts 1.6 p. 23

24 Example (Variational formulation of pure Dirichlet problem for heat equation). BVP: div(σ grad u) = f in, u = g on. (1.6.1) STEP 1 & 2: test with v C0 ( ) div(σ grad u) v dx = Note: v = 0 for test function, because u already fixed on. f v dx. (1.6.2) STEP 3: use Green s formulas on R d (multidimensional integration by parts) Theorem (Green s first formula). If is piecewise smooth, then for all vector fields j (C 1 ( )) 3 and functions v C 1 ( ) div j v + j grad v dx = j n v ds (1.6.3) Proof. [by Gauss theorem ] 1.6 p. 24

25 Apply (1.6.3) to (1.6.2) with j := σ grad u: σ grad u grad v dx σ grad u n v ds = } {{} =0,because v =0 Variational form of (1.6.1): seek u : R, u = g on such that σ grad u grad v dx = f v dx v C0 f v dx v C 0 ( ). ( ). (1.6.4) Example (Variational formulation: heat conduction with general radiation boundary conditions). BVP: div(σ grad u) = f in, σ grad u n = (u) on. (1.6.5) STEP 1: u not fixed test with v C ( ) div(σ grad u) v dx = f v dx v C ( ). STEP 2: apply Green s first formula (1.6.3) σ grad u grad v dx + } σ grad {{ u n } = (u) v ds = f v dx v C ( ). 1.6 p. 25

26 Variational formulation of (1.6.5): seek u : R such that σ grad u grad v dx + (u) v ds = f v dx v C ( ). (1.6.6) Variational form of Neumann problem recovered for (u) = h. Observation: when we test (1.6.6) with v 1 h ds = This is a compatibility condition for the existence of (variational) solutions of the Neumann problem! [In heat conduction: consequence of conservation of energy (1.2.4)] f dx (1.6.7) (1.6.4) & (1.6.6) represent the variational interpretation of the boundary value problems. Theorem If σ is smooth, classical solutions u C 0 ( ) C 2 ( ) of the boundary value problems (1.6.1) and (1.6.5) are also variational solutions. 1.6 p. 26

27 Proof. Apply Theorem as in the derivation of the weak formulations. Theorem Solutions u of (1.6.4) and (1.6.6), respectively, provide weak solutions of the heat equation div(σ grad u) = f in. Proof. Straightforward from Definition Functional framework (1.6.4) and (1.6.6) for (u) = qu (convective cooling, Ex ) or (u) = h (Neumann boundary conditions) are linear variational problems of the form u V : a(u, v) = f (v) v V (1.7.1) where V = real vector space of functions R (trial and test space) a = (symmetric) bilinear form a : V V R 1.7 f = linear form f : V R p. 27

28 Reminder: Definition Given an R-vector space V, a linear form f is a mapping f : V R that satisfies f (αv + βu) = α f (u) + β f (v) u, v V, α, β R. A bilinear form a on V is a mapping a : V V R, for which a(α 1 v 1 + β 1 u 1, α 2 v 2 + β 2 u 2 ) = = α 1 α 2 a(v 1, v 2 ) + α 1 β 2 a(v 1, u 2 ) + β 1 α 2 a(u 1, v 2 ) + β 1 β 2 a(u 1, u 2 ) for all u i, v i V, α i, β i R, i = 1, 2. Reminder: Definition (Norm). A norm V on an R-vector space V is a mapping V : V R + 0, such that v V = 0 v = 0 v V (N1) λv V = λ v V λ R, v V, (N2) w + v V w V + v V w, v V. (N3) 1.7 p. 28

29 vector space + norm normed vector space Definition (Inner product). A bilinear form a on a R-vector space V is called an inner product, if it is symmetric positive definite (s.p.d.), that is, a(u, v) = a(v, u) u, v V, (IP1) v 0 a(v, v) > 0 v V. (IP2) Definition An R-Hilbert space V is a complete, normed vector space, whose norm V is derived from an inner product (, ) V : V V R according to v 2 V := (v, v) V v V. Important properties of (bi-)linear forms: Definition (Continuity of linear forms). A linear form f on a Hilbert space V is continuous, if C f > 0: f (v) C f v V v V. 1.7 p. 29

30 Definition (Continuity of bilinear forms). A bilinear form a on a Hilbert space V is continuous, if C A > 0: a(u, v) C A u V v V u, v V. Definition (Ellipticity of bilinear forms). A bilinear form a on a Hilbert space V is V - elliptic if γ > 0: a(u, u) γ u 2 V u V. γ is called the ellipticity constant. 1.7 p. 30

31 Theorem (Lax-Milgram Lemma). Assume that a and f are a continuous bilinear/linear form on the Hilbert space V. If, moreover, a is V -elliptic, with ellipticity constant γ > 0, then the linear variational problem u V : a(u, v) = f (v) v V (1.7.1) has a unique solution u V that satisfies the stability estimate u V 1 γ sup v V \{0} f (v) v V. (1.7.2) Proof. 1. Uniqueness of solutions: if both u 1, u 2 V solve (1.7.1), then 0 = a(u 1 u 2, v) v V u 1 u 2 2 V 1 γ a(u 1 u 2, u 1 u 2 ) = 0 u 1 = u Existence of solutions: profound functional analysis (requires completeness of V ) 3. Stability of solutions: γ u 2 V a(u, u) = f (u) sup v V \{0} f (v) v V u V. 1.7 p. 31

32 Q: What is V for varational formulations (1.6.4) and (1.6.6)? For (1.6.4): a(u, v) := σ grad u grad v dx a = symmetric bilinear form on C0 ( ) Idea: use a(, ) as inner product on C 0 ( ) energy norm A Continuity/V -ellipticity of a(, ) trivially satisfied a: bilinear (ok) symmetric (ok) positive definite (?) Estimate in 1D for =]a, b[, a, b R: b a u(x) 2 dx 1 2σ (b a)2 a(u, v) = Generalization to R d : b a b a σ (x) u (x) 2 dx u C 0 (]a, b[). σ (x)u (x) v (x) dx symmetric positive definite on C 0 (]a, b[). Theorem (First Poincaré-Friedrichs inequality). If R d, d N, is bounded, then u 0 diam( ) grad u 0 u C 0 ( ). p

33 Notations: v 2 0 := v 2 dx ( L 2 -norm ), diam( ) := sup{ x y, x, y }. Special energy norm: u 2 1 := u grad u 2 0 Assumption (1.2.2) a(u, v) := σ (x) grad u grad v dx C 0 ( ). is inner product on Example =]0, 1[, σ 1 PSfrag replacements { Hat function u(x) = 2x for 0 < x < 1/2, 2(1 x) for 1/2 < x < 1. Compute a(u, u) = 1 0 u u (x) 2 dx = 4 <. 1/2 x 1 u : R, u(0) = u(1) = 0, a(u, u) <, but u C0 ( )! [Meaningful solutions for temperature distribution] 1.7 C0 ( ) not complete w.r.t. norm u A := a(u, u) 1 /2 (C0 ( ), A) no Hilbert space! p. 33

34 Use trick from functional analysis: normed vector space completion completion complete vector space Use V = H 1 0 ( ) := {v : R: v A <, v = 0} Sobolev space = Hilbert space with norm A Notation: H 1 0 ( ) superscript 1, because first derivatives occur in norm subscript 0, because zero on Another important space: L 2 ( ) := {v : R: v 2 0 := v 2 dx < } Notation: L 2 ( ) superscript 2, because square in the definition of norm 0 Note: Discontinuities allowed for functions in L 2 ( ) 1.7 p. 34

35 Do not be afraid of Sobolev spaces! It is only the norms that matter, the spaces are irrelevant! Summary (relationships of concepts): Variational problem norm (ensuring continuity/ellipticity) [ Sobolev space ] Now: Neumann problem for div(σ grad u) = f : variational formulations (1.6.6) a(u, v) = σ grad u grad v dx u, v C ( ). Same bilinear form as for Dirichlet problem, but different space! Obvious: a(u, u) = 0 for u const. on a(, ) is no inner product on C ( ). Estimate in 1D for =]a, b[, a, b R: for u C ( ) u(x) = u(y) + y x u (t) dt, x, y ]a, b[ u(x) = 1 b a b a { y u(y) + x } u (t) dt dy. 1.7 p. 35

36 b a u(x) 2 dx = 1 (b a) 2 2 (b a) 2 2 b a 2 b a 2 b a where at we used (x + y) 2 2(x 2 + y 2 ), x, y R, at we used the Cauchy-Schwarz inequality b b b y 2 u(y) dy + u (t) dt dy dx a a a x b b 2 b b y 2 u(y) dy dx + u (t) dt dy dx a a a a x b 2 2 b b y u(y) dy + a (b a) 2 (b a) u 2 (t) dt dy dx a a x b 2 b b y u(y) dy + y x u (t) 2 dt dy dx a a a x b 2 b u(y) dy + 2(b a) 2 u (t) 2 dt, a a ( b a f (x)g(x) dx) 2 b a f (x) 2 dx b a g(x) 2 dx f, g L 2 (]a, b[) 1.7 p. 36

37 b a b b and u (t) 2 dt = 0 a a Restrict variational problem (1.6.6) to functions with vanishing mean. u(y) dy = 0 Then a(u, u) = inner product u(x) 2 dx = 0 u = 0. Generalization to R d : Theorem (Second Poincaré-Friedrichs inequality). If R d is bounded, then there is C PF = C PF ( ) > 0 such that u 0 C PF grad u 0 u C ( ) := {v C ( ), v(x) dx = 0}. a(, ) is an innner product on C ( ). Appropriate Sobolev space for Neumann problem: H 1 ( ) := {v : R: v dx = 0, v A < }. (1.7.3) = Hilbert space with norm A. }{{} 1.7 p. 37

38 Variational formulation of homogeneous Neumann problem for div(σ grad u) = f : seek u H 1 ( ) such that a(u, v) := σ (x) grad u grad v dx = f (v) := f v dx v H 1 ( ). (1.7.4) Corollary If f L 2 ( ), then (1.7.4) has a unique solution u H 1 ( ) fulfilling u A C 2 PF (σ ) 1 f 0 with C PF from Thm and σ from (1.2.2). Proof. Use the general Cauchy-Schwarz inequality ( 2 f (x) g(x) dx) f (x) 2 dx which implies the continuity of source functional f on H 1 ( ), the estimates v 0 C PF grad v 0, v A σ grad v 0, g(x) 2 dx f, g L 2 ( ), (1.7.5) and sup v H 1 ( )\{0} f (v) v A sup v H 1 ( )\{0} Then, the assertion follows from Lax-Milgram Lemma Thm v 0 v A f 0 C PF σ f p. 38

39 Remark Point source : f = δ p, p f (v) = v(p)! v H0 1 ( ) v(p) R not continuous! Example: u(x) = log log( x /e) on := {x R 2 : x < 1} u A < u = 0 u unbounded Point source meaningless in context of variational interpretation 1.8 Essential and natural boundary conditions Issue: Linear variational problem (1.7.1) for Dirichlet problem? (1.6.4) in Sobolev framework: seek u H0 1 ( ) + g with σ (x) grad u grad v dx = f v dx v H0 1 ( ), (1.8.1) g = extension of Dirichlet data g: g : R, g = g 1.8 p. 39

40 Dirichlet boundary conditions for (1.2.6) directly imposed on test/trial space in (1.8.1): essential boundary conditions for (1.8.1) Yet, (1.8.1) does not match (1.7.1), because H0 1 ( ) + g is affine space. Converting (1.8.1) into a linear variational problem: seek u 0 H0 1 ( ) such that σ (x) grad(u 0 + g) grad v dx = f v dx v H0 1 ( ). (1.8.2) From u 0 we recover: u = u 0 + g. (1.8.2) = linear variational problem (1.7.1) with a(u, v) = σ (x) grad u grad v dx (see (1.6.4)), f (v) = f v dx σ (x) grad g grad v dx. } {{} Has to be continuous on H0 1( ) Continuity of source term requirement on g 1.8 Demand g 2 1 := g grad g 2 0 < p. 40

41 g H 1 ( ) := {v : R: v 1 < } Continuity of f : H0 1 ( ) R by Cauchy-Schwarz inequality (1.7.5) Remark. If g : R piecewise smooth, but discontinuous g What are valid Dirichlet data g? [Jump in temperature on would entail infinitely thin perfect insulator] 1. Piecewise smooth, continuous functions on 2. Functions belonging to the trace space ( R d ) g H 1 /2 ( ) := {v : R: v 2 1/2, := v 2 ds + v(x) v(y) 2 x y d+1 ds(x, y) < }. 1.8 p. 41

42 Theorem (Trace theorem for H 1 ( )). The pointwise restriction mapping γ : C ( ) C 0 ( ), (γ u)(x) = u(x) for all x, satisfies C = C( ) > 0: γ u 1 /2, C u 1 u C ( ), and, hence, can be extended to a continuous trace operator γ : H 1 ( ) H 1 /2 ( ), that has a continuous right inverse (an extension operator E : H 1 /2 ( ) H 1 ( )). Extend (1.7.4): variational formulation of inhomogeneous Neumann problem: seek u H 1( ) σ (x) grad u grad v dx = f v dx + hv ds v H 1 ( ). (1.8.3) Linear variational problem according to (1.7.1) Neumann boundary conditions for (1.2.6) do not show up in trial/test space: natural boundary conditions for (1.8.3) Q: Requirements on h to get continuous v H 1( ) 1.8 hv ds? p. 42

43 Cauchy-Schwarz inequality (1.7.5) on + Trace theorem Remark Are Neumann boundary conditions contained in (1.8.3)? h L 2 ( Ɣ) is enough! Test with v C0 ( ) and Thm : div(σ grad u) = f in Then test with v C ( ), use PDE, and Thm : σ grad u n = h on }{{} PDE + boundary conditions What about mixed boundary conditions: = Ɣ D Ɣ N, Ɣ D, Ɣ N Use V = H 1 Ɣ D ( ) := {v : R: v A <, v ƔD = 0} Generalization of 1st Poincaré-Friedrichs inequality, Thm : Theorem (General Poincaré-Friedrichs inequality). There is C > 0 depending on, Ɣ D such that u 0 C grad u 0 u H 1 Ɣ D ( ). What about pure convective cooling conditions ( Ex ) j n = q(u u 0 ) in (1.6.6)? 1.8 p. 43

44 Variational formulation: seek u H 1 ( ) such that σ (x) grad u grad v dx + qu v ds = f v dx + Trace theorem Cauchy-Schwarz inequality (1.7.5) qu 0 v ds v H 1 ( ). continuity of bilinear form/source functional H 1 ( )-ellipticity of bilinear form exercises 1.9 The Dirichlet principle Linear variational problem u V : a(u, v) = f (v) v V, (1.7.1) V = Hilbert space with norm V, a(, ) continuous bilinear form, f continuous linear form. Assumption: V -ellipticity of a(, ), see Def : γ > 0: a(u, u) γ u 2 V u V. (1.9.1) p

45 Assumption: a(, ) symmetric: a(u, v) = a(v, u) u, v V Theorem (Dirichlet principle). Assuming (1.9.1), u V solves (1.7.1), if and only if it is the unique solution of u = arg min J(v), J(v) := 1 v V 2a(v, v) f (v). Proof. Lax-Milgram lemma Thm existence & uniqueness of solution u of (1.7.1). Then for all v V, using a(u, v) = f (v), J(v) J(u) = 1 2 (a(v, v) a(u, u)) f (v u) = 2 1 a(v, v) 2 1 a(u, u) a(u, v u) (1.9.2) = 2 1 a(v u, v u) 2 1 γ u v 2 V. J(v) > J(u) and ( J(v) = J(u) u = v). Conversely, (1.9.2) u is unique solution of minimization problem for J. 1.9 p. 45

46 The Dirichlet principle The variational formulations (1.8.2), (1.8.3) of linear scalar second-order elliptic boundary value problems are equivalent to minimization problems for quadratic functionals (also known as Dirichlet forms or energy functionals) Example (Quadratic functionals). Analogy parabola: J(v) = 1 2a(v, v) f (v) f (x) = ax 2 + bx quadratic functional R 2 R Fig p. 46

47 2 The Finite Element Method (FEM) Problem : scalar second-order elliptic boundary value problem Perspective : variational interpretation in Sobolev spaces Objective : algorithm for the computation of an approximate numerical solution 2.1 Fundamentals Moot point: any computer can only handle a finite amount of information (reals) Variational boundary value problem DISCRETIZATION System of a finite number of equations for real unknowns 2.1 p. 47

48 2.1.1 Galerkin discretization Abstract discussion: start from linear variational problem (see Sect. 1.7, (1.7.1)) u V : a(u, v) = f (v) v V, (1.7.1) V = Hilbert space with norm V, a(, ) continuous bilinear form, f continuous linear form. Norm of a(, ): C A := sup v V \{0} sup u V \{0} a(u, v) u V v V < Assumption: V -ellipticity of a(, ), see Def : γ > 0: a(u, u) γ u 2 V u V. (1.9.1) Remark. If a(, ) symmetric ( inner product, see Def ) and V = energy norm A γ, C A = 1 Idea of Galerkin discretization Replace V in (1.7.1) with a finite dimensional subspace V N (discrete trial/test space). 2.1 p. 48

49 Notation: N = formal index, tagging discrete entities ( finite amount of information ) Discrete variational problem u N V N : a(u N, v N ) = f (v N ) v N V N. (2.1.1) Lax-Milgram Lemma Thm Existence & Uniqueness of solution u N V N, stability u N V 1 γ f (v N ) sup. v N V N \{0} v N V Issues: 1. How accurate is the Galerkin solution u N? (a) What measure for accuracy? (b) How to assess accuracy? 2. How to convert (2.1.1) into (linear) system of equations? Ad 1(a): Focus on norm V (and A, if a(, ) inner product) 2.1 p. 49

50 Galerkin orthogonality a(u u N, v N ) = 0 v N V N. (2.1.2) [Geometric meaning for inner product a(, ) ] V u e N := u u N V N u N Fig. 6 Discretization error e N := u u N a(, )-orthogonal to discrete trial/test space V N Theorem (Cea s lemma). If a(, ) = continuous, V -elliptic, bilinear form, V N V finite dimensional subspace, u V / u N V N solve (1.7.1)/ (2.1.1), then u u N V C A γ inf v N V N \{0} u v N V Proof. (of a weaker assertion, for complete proof see J. XU AND L. ZIKATANOV, Some oberservations on Babuška and Brezzi theories, Numer. Math. 2003) 2.1 p. 50

51 By Galerkin-orthogonality (2.1.2), for all v N V N because v N arbitrary. γ u u N 2 V a(u u N, u u N )+a(u u N, u N v N ) = a(u u N, u v N ) C A u u N V u v N V. u u N V C A γ inf v N V N \{0} u v N V, Quasi-optimality of Galerkin solutions: with C > 0 independent of u, V N u u N V } {{ } C (norm of) discretization error inf v N V N u v N V }{{} best approximation error, To assess accuracy of Galerkin solution: study capability of V N to approximate u! Monotonicity of Galerkin schemes Trial test spaces V N, V N V : V N V N u u N V u u N V. Enhance accuracy by enlarging ( refining ) trial space. 2.1 p. 51

52 Remark If a(, ) is inner product on V : Phythagoras theorem Fig. 6 u u N 2 A = u 2 A u N 2 A. (2.1.3) Reminder: Definition (Linear operator). Let V, W be real vector spaces. A mapping T : V W is a linear operator, if T(αu + βv) = αt(u) + βt(v) u, v V, α, β R. Reminder: Definition (projection). projection, if P 2 = P. A linear operator P : V V on a vector space V is a 2.1 p. 52

53 Definition (Galerkin projection). Under the assumptions of Thm the Galerkin projection P N : V V N V is defined by a(p N u, v N ) = a(u, v N ) v N V N. [Lax-Milgram Lemma Thm P N well defined and continuous] The (linear) algebraic setting [Now we tackle issue 2. I. (conversion of (2.1.1) into system of equations)] Introduce (ordered) basis B N of V N : B N := {b 1 N,..., bn N } V N, V N = Span {B N }, N := dim(v N ). II. Basis representations u N = µ 1 b 1 N + + µ N b N N, v N = ν 1 b 1 N + + ν Nb N N, µ i R ν i R in (2.1.1). 2.1 (2.1.1): a(u N, v N ) = f (v N ) v N V N p. 53

54 a(µ 1 b 1 N + + µ N b N N, ν 1b 1 N + + ν N b N N ) = f (ν 1b 1 N + + ν N b N N ) ν 1,..., ν N R, N k=1 N µ k ν j a(b k N, b j N N ) = ν j f (b j N ) ν 1,..., ν N R, j=1 j=1 N k=1 µ ka(b k N, b j N ) = f (b j N ) for j = 1,..., N. A µ = ϕ, A = ( a(b k N, b j N ) ) N j,k=1 RN,N, ϕ = ( f (b j ) N N ), j=1 µ = (µ 1,..., µ N ) T R N 2.1 p. 54

55 Discrete variational problem u N V N : a(u N, v N ) = f (v N ) v N V N Choosing basis B N ( Stiffness matrix: A = a(b k N, b j ) N N ) j,k=1 RN,N, ( Load vector: ϕ = f (b j ) N N ) j=1 RN, Coefficient vector: µ = (µ 1,..., µ N ) T R N, u N = N k=1 µ k b k N. Recovery of solution: Linear system of equations A µ = ϕ Corollary (2.1.1) has unique solution A regular Impact of choice of basis? Choice of B N does not affect u N No impact on discretization error! 2.1 p. 55

56 Properties of matrix A crucially depend on basis B N! Lemma Consider (2.1.1) and two bases of V N, B N := {b 1 N,..., bn N }, B N := {b1 N,..., bn N }, related by b j N N = s jk b k N with S = (s jk) N j,k=1 RN,N regular. k=1 Stiffness matrices A, A R N,N, load vectors ϕ, ϕ R N, and coefficient vectors µ, µ R N, respectively, satisfy A = SAS T, ϕ = S ϕ, µ = S T µ. (2.1.4) Proof. A lm = a(b m N, bl N ) = N k=1 N s mk a(b k N, b j N )s l j = j=1 N ( N ) s l j A jk s mk = (SAS T ) lm, k=1 j=1 } {{ } (SA) lk 2.1 p. 56

57 Reminder of linear algebra: Definition (Congruent matrices). Two matrices A R N,N, B R N,N, N N, are called congruent, if there is a regular matrix S R N,N such that B = SAS T. Equivalence relation on square matrices Lemma Matrix property invariant under congruence Property of stiffness matrix invariant under change of basis B N Matrix properties invariant under congruence = regularity symmetry positive definiteness Reminder: Definition (Positive definite matrix). Matrix B R N,N, N N, is positive definite ξ T B ξ > 0 for all ξ R N \ {0}. 2.1 p. 57

58 2.1.3 Principles of FEM R d, d = 2, 3, bounded computational domain: assumed polygonal d = 2, polyhedral d = 3 First main ingredient: triangulation/mesh of Definition A mesh (or triangulation) of R d is a finite collection {K i } i=1 M, M N, of open non-degenerate polygons (d = 2)/polyhedra (d = 3) such that (A) = {K i, i = 1,..., M}, (B) K i K j = i j, (C) for all i, j {1,..., M}, i j, the intersection K i K j is a vertex, edge, or face of both K i and K j. vertex, edge, face of polygon/polyhedron: geometric intuition Terminology: Given mesh M := {K i } M i=1 : K i called cell or element. Vertices of a mesh nodes (set N (M)) 2.1 p. 58

59 Types of meshes: Triangular mesh in 2D Fig. 7 Quadrilateral mesh in 2D Fig. 8 If (C) does not hold Triangular non-conforming mesh (with hanging nodes) K i K j is only part of an edge/face for at most one of the adjacent cells. (However, conforming if degenerate quadrilaterals admitted) Fig. 9 Simplicial mesh = triangular mesh in 2D 2.1 tetrahedral mesh in 3D p. 59

60 Second main ingredient: space of piecewise polynomial functions V N := {v V : v K P p (K ) K M}, P p (K ) = polynomials of degree p on cell K. Note: v V conformity conditions at interelement boundaries Lemma (Conformity condition for H 1 ). Let M := {K i } i=1 M be a triangulation ( Def ) of R d and assume that v : R satisfies that v K can be extended to a function in C (K ) for any K M. Then v H 1 ( ) v C 0 ( ). Conformity condition for H 1 = global continuity (C 0, not C 1! Ex ) (recall physical constraints on temperature distributions!) Thanks to notion of weak derivative, Sect. 1.5! 2.1 p. 60

61 Definition (Conformity). A M-piecewise polynomial space V N is called V -conforming, if V N V. Third main ingredient: Locally supported basis functions Basis functions b 1 N,..., bn N for a finite element trial/test space V N built on a mesh M satisfy: each b i N associated with a single cell/edge/face/vertex of M, supp(b i N ) = {K : K M, p K }, if b i N associated with cell/edge/face/vertex p. Finite element terminology: b i N = global shape/basis functions Example (Supports of global shape functions in 1D). 2.1 p. 61

62 =]a, b[ ˆ= interval Equidistant mesh Support of global shape function associated with x 7 M := {]x j 1, x j [, j = 1,..., N}, x j := a + hj, h := (b a)/n, N N. 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 Example (Supports of global shape functions on triangular mesh). Fig. 10 Support of node-associated basis function Fig. 11 Support of edge-associated basis function Fig. 12 Support of cell-associated basis function Rationale for small supports? 2.1 p. 62

63 Recall bilinear form : a(u, v) := grad u grad v dx Use triangular mesh M, test/trial space V N H 1 ( ) with basis B N := {b 1 N,..., bn N } ( Sect ) Stiffness matrix A R N,N with a i j := a(b j N, bi N ), i, j = 1,..., N b i N, b j N associated with nodes not linked by an edge a i j = 0 (because vol(supp(b i N ) supp(b j N )) = 0) Finite element stiffness matrices are sparse. Definition (Sparse matrix). A matrix A R N,N is called sparse, if nnz(a) := {(i, j): a i j 0} N p. 63

64 Example (Sparse stiffness matrices). V N : one basis function associated with each vertex Finite element mesh M nz = 2670 Resulting sparsity pattern of stiffness matrix Visualization of sparsity pattern: MATLAB-spy()-Funktion 2.1 p. 64

65 Remark (Storing sparse matrices). Special (efficient) storage formats for sparse matrices, e.g., CRS-format Special MATLAB commands: ( use mandatory!) sparse, spones, speye, spdiags Sect.?? Matrix-Speicherformate, course Numerische Mathematik für CSE Linear H 1 -conforming finite elements M = simplicial mesh of polygonal/polyhedral computational domain R d, d = 2, 3 Linear H 1 -conforming finite elements = Simplest H 1 ( )-conforming finite element space = Simplest finite element scheme for scalar second order elliptic BVP on p

66 S 0 1 (M) := {v C0 ( ): v K P 1 (K ) K M} H 1 ( ) Representation: P 1 (K ) := {x α + β x, x K, α R, β R d }. (space of d-variate polynomials of total degree 1) dim P 1 (K ) = d + 1 Notation: S 0 1 (M) continuous functions, cf. C 0 ( ) locally 1st degree polynomials, cf. P 1 Example (H 1 ( )-conforming linear finite element space in 1D) d = 1, =]0, 1[, mesh M = partion of ]0, 1[ into intervals 1 red: function S1 0(M) blue: hat function basis of S1 0(M) Locally supported basis functions in 2D? p. 66

67 On a triangle T with vertices a 1, a 2, a 3 : q P 1 (T ) uniquely determined by values q(a i ). v N S1 0 (M) uniquely determined by {v(x), x node of M}! dim S1 0 (M) = V(M) (V(M) = set of vertices of M) If V(M) = {x 1,..., x N }, nodal basis B N := {b 1 N,..., bn N } of S0 1 (M) defined by bi N (x j ) = δ i j. Piecewise linear nodal basis function ( hat function ) (= global shape function for S 0 1 (M)) 1 coefficient µ j = nodal value of u N at j-th node of M Global shape functions Restriction to element local shape functions (2.1.5) 2.1 p. 67

68 Example (Local shape functions for S 0 1 (M)). Triangle K with vertices a 1 = ( 0 0 ), a 2 = ( 1 0 ), a 3 = ( 0 1 ) : Local shape functions: b 1 K (x) = 1 x 1 x 2, b 2 K (x) = x 1, b 3 K (x) = x 2. Fig. 13 Local shape functions for S1 0 (M) on triangle/tetrahedron = barycentric coordinate functions Definition (Barycentric coordinates). Given d + 1 points a 1,..., a d+1 R d that do not lie in a hyperplane the barycentric coordinates λ 1 = λ 1 (x),..., λ d+1 = λ d+1 (x) R of x R d are uniquely defined by λ 1 (x) + + λ d+1 (x) = 1, λ 1 (x) a λ d+1 (x)a d+1 = x x R d. 2.1 p. 68

69 Barycentric coordinates obtained by solving a 1 1 ad ad 1 ad+1 d 1 1 λ 1. λ d λ d+1 = x 1. x d 1. (2.1.6) Corollary Given d + 1 points a 1,..., a d+1 R d as in Def , the barycentric coordinates are affine linear functions on R d, which satisfy { λ j (a i 1, if i = j, ) = δ i j := 1 i, j d else, 2.1 p. 69

70 Fig. 14 Barycentric coordinate functions (= Local shape functions for S1 0 (M)) on a triangle How to get H 1 0 ( )-conforming finite element space S0 1,0 (M) := S0 1 (M) H 1 0 ( )? Discard nodal basis functions associated with vertices on! Remark Piecewise linear finite element subspace of H 1 ( )? There exist no locally supported piecewise linear basis functions. 2.1 p. 70

71 2.1.5 Simplicial Lagrangian finite elements M = simplicial mesh of polygonal/polyhedral computational domain R d, d = 2, 3 Idea: Use higher degree polynomials better accuracy (cf. interpolation) Higher degree polynomials P p (R d ) := {x R d α N d 0, α p κ αx α, κ α R}. Notation: α = multiindex (α 1,..., α d ), α = α α d, x α := x α 1 1 xα d d. Example: } P 2 (R 2 ) = Span {1, x 1, x 2, x1 2, x2 2, x 1x 2 Lemma dim P p (R d ) = ( ) d + p p for all p N, d N Definition (Higher order Lagrangian finite element spaces). Space of p-th degree Lagrangian finite element functions on mesh M S 0 p (M) := {v C0 ( ): v K P p (K ) K M}. 2.1 p. 71

72 Notation: S 0 p (M) continuous functions, cf. C 0 ( ) locally polynomials of degree p, cf. P p (R d ) Construction: Local shape functions Glueing Global FE space (Glueing must ensure global continuity H 1 ( )-conformity)! Design of local shape functions must make glueing possible Example (Quadratic Lagrangian finite elements). Local shape functions for P 2 (K ), K triangle: b1 K = λ 1(1 2λ 1 ), b2 K = λ 2(1 2λ 2 ), b12 K = 4λ 1λ 2, b13 K PSfrag replacements = 4λ 1λ 3, b3 K = λ 3(1 2λ 3 ), b23 K = 4λ 2λ 3. λ i = barycentric coordinate function ( Def ) for vertex a i a 3 a 13 a 23 b K j (a i) = δ i j, i, j {1, 2, 3, (12), (23), (13)}. a 2 a a p. 72

73 Fig. 15 Local shape functions = Lagrangian (interpolatory) polynomials for local nodes in K Specifying local interpolation nodes specifying local shape functions When are local nodes q i, i = 1,..., Q, (for P p (K )) suitable for glueing? Unisolvence: q Pp (K ): v(q i ) = 0 i v 0 Fixing traces: locally unisolvent interpolation on each vertex/edge/face 2.1 p. 73

74 Invalid choice of local nodes for S 0 2 (M) Interelement matching: corresponding nodes on joint edges/faces Fig. 16 Matching nodes for quadratic Lagrangian finite elements 2.1 p. 74

75 Fig. 17 Glueing : edge-associated local and resulting global shape function for S2 0 (M), M triangular Fig. 18 Fig. 19 Fig. 20 Fig. 21 Location of local (interpolation) nodes for triangular Lagrangian finite elements of degree 2 (left), degree 3 (middle), and degree 4 (right) 2.1 p. 75

76 Fig. 22 Fig. 23 Fig. 24 Local nodes for tetrahedral Lagrangian finite elements (left: p = 2, middle: p = 3, right: p = 4) Can we find other locally supported bases for S 0 2 (M)? YES! Alternative p-hierarchical local shape functions: b K 1 = λ 1, b K 12 = 4λ 1λ 2, b K 2 = λ 2, b K 13 = 4λ 1λ 3, b K 3 = λ 3, b K 23 = 4λ 2λ 3. Set comprises local shape functions for p = 1. Glueing can easily be accomplished 2.1 p. 76

77 No canoncical local shape functions/global basis functions for higher order Lagrangian finite elements. Selection of B N to get good matrix properties Parametric finite elements Definition (Affine transformation). Mapping : R d R d affine, if (x) = Fx + τ with F R d,d, τ R d. Usually: All elements of a mesh = affine images of reference element(s) K K K M: F K R d,d regular, τ K R d : K = K ( K ) with K ( x) := F K x + τ K. 2.1 p. 77

78 (00 ) ( Unit triangle : K =, 10 ) (, 01 ) { For K = convex a 1, a 2, a 3} : ( ) 1 x 2 K K a 3 F K = a 2 1 a1 1 a3 1 a1 1 a 2 2 a1 2 a3 2 a1 2, τ K = a 1. K a 1 1 x 1 a 2 Fig. 25 Transformations of elements transformation of functions: Definition (Pullback). Given domains, and a bijective mapping :, the pullback u : R of a function u : R is defined by ( u)( x) := u( ( x)) x. Notation: If unambiguous: û := u 2.1 p. 78

79 Note: Consider S1 0(M), triangle K M, unit triangle K, affine mapping K : K K b1 K, bk 2, bk 3 (standard) local shape functions on K b 1, b 2, b 3 (standard) local shape functions on K Observation: b i = K bk i Terminology: affine equivalent finite elements All families of Lagrangian finite elements (Sect ) (equipped with natural local shape functions) are affine equivalent. STEP 1: define local shape functions on reference element K STEP 2: local shape functions on K M via pullback ( 1 ) ( Def ) Parametric finite elements Generalization: curvilinear meshes with curved edges/faces ( Sect ) 2.1 p. 79

80 Now: K diffeomorphism! b K i := ( 1 K ) b i. 1 x 2 K K a 3 K a 1 1 x 1 a 2 Fig. 26 Application: approximation of curved interfaces/boundaries ( Sect 2.2.8) Lagrangian finite elements on quadrilaterals/hexahedra Parametric construction: start from reference element K =]0, 1[ d (unit cube) Lowest polynomial degree p = 1, 2D: piecewise bilinear finite elements 2.1 p. 80

81 Local shape functions on K =]0, 1[ 2 PSfrag replacements b 1 = (1 x 1 )(1 x 2 ), b 2 = x 1 (1 x 2 ), b 3 = x 1 x 2, b 4 = (1 x 1 ) x 2. x 2 a 4 a 3 Bilinear Lagrangian interpolation polynomials w.r.t. points a i of K corner Note: b i linear on edges a 1 a 2 x 1 Fig. 27 Bilinear local shape functions on unit square K 2.1 p. 81

82 General quadrilateral x 2 a 4 Unit square a 3 Parallelogram K K Bi linear mapping a 1 a 2 x 1 Affine mapping Bilinear mapping to general quadrilateral: ( ) α1 + β K ( x) = 1 x 1 + γ 1 x 2 + δ 1 x 1 x 2 α 2 + β 2 x 2 + γ 2 x 2 + δ 2 x 2 x 2, α i, β i, γ i, δ i R. b i, i = 1, 2, 3, 4, bilinear local shape functions on ]0, 1[ 2, K : K K bilinear Example (Triangle as degenerate quadrilateral). bk i = ( 1 K ) b i linear on edges of K 2.1 p. 82

83 x 2 x 2 ts K K ( x) = ( x 1 1 ) 2 x 1 x 2 x x 1 x 2. x 1 x 1 Higher order quadrilateral Lagrangian finite elements: Definition (Tensor product polynomials). Space of tensor product polynomials of degree p N in each coordinate direction Q p (R d ) := {x p 1 (x 1 ) p d (x d ), p i P p (R), i = 1,..., d}. Local trial space Q p ( K ) on K =]0, 1[ 2 + parametric construction p

84 ts x 2 a 4 PSfrag replacements a 3 x 2 a 4 PSfrag replacements a 3 x 2 a 4 a 3 Fig. 28 a 1 a 2 Nodal points p = 1 x 1 a 1 a 2 Nodal points p = 2 x 1 a 1 a 2 Nodal points p = 3 x Degrees of freedom Recall: Lagrangian local shape functions bi K Q N ( Sects , 2.1.7). fixed by b K i (q j ) = δ i j for nodes q i, i, j = 1,..., Q, Definition (Dual basis). Given a vector space V with basis B := {b 1,..., b Q }, Q N, the corresponding dual basis is a set l 1,..., l Q of linear forms on V such that l j (b i ) = δ i j, i, j {1,..., Q}. 2.1 p. 84

85 For Lagrangian finite elements S 0 p (M), on element K : Nodal evaluation functionals v v(q j ), j = 1,..., Q, form dual basis w.r.t. basis {b K 1,..., bk Q } of local trial space S p(k ). Definition (Local degrees of freedom). A dual basis of the local trial space corresponding to the local shape functions provides local degrees of freedom (d.o.f.). Role reversal: degrees of freedom local shape functions Example (Cubic Hermitian Finite Elements on triangular mesh). Local trial space V K := P 3 (K ) for each K M, Local degrees of freedom, K triangle with vertices a 1, a 2, a 3 R 2 l 1 (v) = v(a 1 ), l 2 (v) = v(a 2 ), l 3 (v) = v(a 3 ), l 4 (v) = grad v(a 1 ) (a 2 a 1 ), l 5 (v) = grad v(a 2 ) (a 3 a 2 ), l 6 (v) = grad v(a 3 ) (a 1 a 3 ), l 7 (v) = grad v(a 1 ) (a 3 a 1 ), l 8 (v) = grad v(a 2 ) (a 1 a 2 ), l 9 (v) = grad v(a 3 ) (a 2 a 3 ), l 10 (v) = v( 3 1 (a1 + a 2 + a 3 )). 2.1 p. 85

86 Dual basis ( Def )? functionals = dim V K = dim P 3 (R 2 ) = ( 3+2) 2 = 10, If l j (v) = 0 for all j = 1,..., 9, then v(a i ) = 0 and grad v(a i ) = 0, i = 1, 2, 3, v P 3 (K ) 0 on any edge, (= 0 at center of gravity) v 0 for any v V K. Unisolvence of local d.o.f. Suitable for glueing? YES, because v edge uniquely determined by d.o.f. associated with the edge. A local degree of freedom l is regarded as associated with an edge E, if l(v) only depends on v E, grad v E,.... Symbolic notation for local d.o.f. for cubic Hermitian elements: (filled circle = nodal values, circle = first derivatives, arrows = directional derivatives) 2.1 Fig. 29 p. 86

87 HOWEVER, alternative choice of local degrees of freedom possible (on triangle K with vertices a 1, a 2, a 3 R 2 ) l 1 (v) = v(a 1 ), l 2 (v) = v(a 2 ), l 3 (v) = v(a 3 ), l 4 (v) = v (a 1 ), x 1 l 5 (v) = v (a 2 ), x 1 l 6 (v) = v (a 3 ), x 1 l 7 (v) = v x 2 (a 1 ), l 10 (v) = v(a 123 ). l 8 (v) = v x 2 (a 2 ), l 9 (v) = v x 2 (a 3 ), Three d.o.f. associated with each vertex Fewer global shape functions compared to previous choice! Fig p. 87

88 2.2 Finite Element Implementation Mesh file format Data flow in (most) finite element software packages: Parameters Mesh generator Finite element solver (computational kernel) Post-processor (e.g. visualization) Example (Mesh file format (triangular mesh of polygonal domain)). 2.2 p. 88

89 # Two-dimensional simplicial mesh 1 ξ 1 η 1 # Coordinates of first node 2 ξ 2 η 2 # Coordinates of second node. N ξ N η N # Coordinates of N-th node 1 n 1 1 n 1 2 n 1 3 X 1 # Indices of nodes of first triangle 2 n 2 1 n 2 2 n 2 3 X 2 # Indices of nodes of second triangle. M n1 M n2 M n3 M X M # Indices of nodes of M-th triangle X i, i = 1,..., M extra information (e.g. material properties in triangle #i). (2.2.1) Optional: additional information about edges (on ): K N # Number of edges on n 1 1 n1 2 Y 1 # Indices of endpoints of first edge n 2 1 n2 2 Y 2 # Indices of endpoints of second edge. n1 K nk 2 Y K # Indices of endpoints of K -th edge (2.2.2) 2.2 p. 89

90 Y k, k = 1,..., K extra information Example (Mesh file format for MATLAB code). Vertex coordinate file: % List of vertices e e e e e e e e e e+00 Cell information file: % List of elements p. 90

91 1 3 Loading a mesh m = load Mesh( Coord Circ.dat,... Elem Circ.dat ); plot Mesh(m, apts ); Option flags: a : with axes p : vertex labels on t : cell labels on s : caption/title on Fig. 31 How to create a mesh? 2.2 p. 91

92 Mesh generation (beyond scope of this course) Free software: NETGEN ( Triangle ( quake/triangle.html) TETGEN ( Example (Mesh generation in MATLAB code). Algorithm & details [6] MATLAB-CODE: mesh generation for circular domain BBOX = [-1-1; 1 1]; H0 = 0.1; DHD sqrt(x(:,1).ˆ2+x(:,2).ˆ2)-1; HHANDLE ones(size(x,1),1); Mesh = init Mesh(BBOX,H0,DHD,... HHANDLE,[],1); save Mesh(Mesh, Coordinates.dat,... Elements.dat ); Bounding box Largest reasonable edge length Signed distance function ϕ(x): (distance from, ϕ(x) < 0 x ) Element size function (determines local edge length) 2.2 p. 92

93 2.2.2 Assembly term used for computing entries of stiffness matrix/load vector. Discrete variational problem (V N = FE space, dim V N = N N, see Sect ) To be computed: u N V N : a(u N, v N ) = f (v N ) v N V N. (2.1.1) Stiffness matrix: A = ( a(b j N, bi N ) ) N i, j=1 RN,N Load vector: ϕ := ( f (b i N ) ) N i=1 RN } {{ } both can be written in terms of cell contributions: a(u, v) = a K (u K, v K ), f (v) = K M K M f K (v K ). (2.2.3) 2.2 p. 93

94 Example: bilinear form/linear form arising from 2nd-order elliptic BVPs ( Sect. 1.6) a(u, v) := σ grad u grad v dx = σ grad u grad v dx, K M } K {{} =:a K (u K,v K ) f (v) := f v dx = f v dx. K M } K {{} =: f K (v K ) Recall (2.1.5): Restrictions of global shape functions to cells = local shape functions Definition Given local shape functions {b1 K,..., bk Q }, we call ( ) element stiffness matrix A K := a K (b K Q j, bk i ) i, j=1 RQ,Q, ( ) Q element load vector ϕ K := f K (bi K ) i=1 RQ. 2.2 p. 94

95 Theorem The stiffness matrix and load vector can be obtained from their cell counterparts by A = K T T K A K T K, ϕ = K T T K ϕ K, (2.2.4) with the index mapping matrices ( T-matrices ) T K R Q,N, defined by { 1, if (b j (T K ) i j := N ) K = bi K, 1 i Q, 1 j N. 0, otherwise. Proof. (A) i j = a(b j N, bi N ) = K M a K (b j N K, bi N K ) = K M, supp(b j N ) K, supp(b i N ) K a K (b K l( j), bk l(i) ) = K M, supp(b j N ) K, supp(b i N ) K (A K ) l(i),l( j). l(i) {1,..., k K }, 1 i N ˆ= index of the local shape function corresponding to the global shape function b i N on K. (A) i j = K M, supp(b j N ) K, supp(b i N ) K Q l=1 n=1 Q (T K ) li (A K ) ln (T K ) nj. 2.2 p. 95

96 Example (Assembly for linear Lagrangian finite elements on triangular mesh) Using the local/global numbering indicated beside T K = PSfrag replacements K Fig Mesh data structures 2.2 p. 96

97 Bare minimum of information a mesh data structure has to provide is 1. a unique identification and the geometric location of global basis functions, 2. a possibility to traverse the local shape functions/degrees of freedom of every cell, 3. a way to run through the edges/faces of a cell in predefined order, 4. and a method for iterating through all cells of the mesh ( global numbering) Focus: array oriented data layout ( MATLAB, FORTRAN) Case: d-dimensional simplicial triangulation M, minimal data structure (cf. Sect ) Coordinates of vertices N (M) : N (M) d-array Coordinates of double Vertex indices for cells: M (d + 1)-array Elements of int. 2.2 p. 97

98 ( 1, 1) (1, 1) ( 1, 1) 4 (1, 1) Example (Arrays storing 2D triangular mesh). i Coordinates Array Coordinates K j Vertex indices Array Elements Fig. 33 Global shape functions associated with edges/faces extra information required! Example (Extended MATLAB mesh data structure). mesh = add Edge2Elem(add Edges(init Mesh(BBOX,H0,DHD,HHANDLE,[],1))) (init Mesh Ex ) 2.2 p. 98

99 mesh = Coordinates: [5x2 double] Elements: [4x3 double] Edges: [8x2 double] Vert2Edge: [5x5 double] Edge2Elem: [8x2 double] EdgeLoc: [8x2 double] vertex coordinates, see Ex vertex indices of triangles, see Ex indices of endpoints in Coordinates array N (M) N (M) sparse integer matrix: entry (i, j) = edge index, if 0 E(M) 2 integer array: indices of adjacent cells in Elements array E(M) 2 integer array: local indices of edges w.r.t. adjacent cells Notation: E(M) ˆ= edges of 2D mesh How to number order local shape functions global shape functions? Elements, Edges arrays ordering of vertices of cells/endpoints of edges Arrays (of vertices,cells,edges) array indices numbering of global shape functions 2.2 p. 99

100 2.2.4 Algorithms Guideline: Cell oriented assembly (2.2.4) Loop: foreach K M do { local operations on K ( A K ) + A = A + T T K A K T K } Notion: local operations ˆ= required only data from fixed neighbourhood of K computational effort O(1) : independent of M Cell oriented assembly in MATLAB function A = assemble(mesh) for k = Mesh.Elements idx = ❶ Aloc = ❷ A(idx,idx) = A(idx,idx)+Aloc; end ❶ row vector of index numbers of global shape functions b i1,..., b iq V N correspondin to local shape functions b K 1,..., bk Q : idx = (i1,..., iq) (encodes index mapping matrix T K ) ❷ Q Q element stiffness matrix For Lagrangian FEM ( Sect ): the total computational effort is of the order O( M) = O(N), N := dim V N. p

101 Example (Assembly for quadratic Lagrangian FE in MATLAB code). Setting: FE space S2 0 (M) on triangular mesh M of polygon R2 Recall: 6 local shape functions: 3 vertex-associated, 3 edge-associated Ex , Sect Convention: vertex-associated global shape functions b 1,..., b M edge-associated global shape PSfrag functions replacements b M+1,..., b M+ E(M) 3 Local numbering p. 101

102 MATLAB-CODE: assembly for quadratic Lagrangian FE function A = assemmat QFE(Mesh,EHa 1 ndle,varargin) nv = size(mesh.coordinates,1); ne = size(mesh.elements,1) I 2 = zeros(36*ne,1); J = I; a = I; offset = 0; for k =1:nE vidx = Mesh.Elements(k) idx 3 = [vidx,... Mesh.Vert2Edge(vidx(1),vidx(2))+nV,... Mesh.Vert2Edge(vidx(2),vidx(3))+nV,... Mesh.Vert2Edge(vidx(3),vidx(1))+nV]; Aloc 4 = transpose(ehandle(mesh.coordinates(vidx,:),... Mesh. 5 ElemFlag(k),varargin{:})); Qsq = prod(size(aloc)); range = offset + 1:Qsq; t = idx(ones(length(idx),1),:) ; I(range) = t(:); t = idx(ones(1,length(idx)),:); J(range) = t(:); a(range) = Aloc(:); offset = offset + Qsq; end A 6 = sparse(i,j,a); 2.2 p. 102

103 1: EHandle (function handle) provides element stiffness matrix A K R 6,6 2: I,J,a ˆ= linear arrays storing (i, j, a i j ) for stiffness matrix A. Initialized with 0 for the sake of efficiency Ex : idx ˆ= index mapping vector, see ❶ above 4: Aloc = A K R 6,6 (element stiffness matrix) 5: Mesh.ElemFlag(k) marks groups of elements (e.g. to select local heat conductivity σ in (1.6.4)) 6: Build sparse MATLAB-matrix ( Def ) from index-entry arrays Example (Efficient implementation of assembly). tic-toc-timing (min of 4v runs), MATLAB V7, Intel Pentium 4 Mobile CPU 1.80GHz, Linux Computation of element stiffness matrices skipped! 2.2 p. 103

104 Sparse assembly: A(idx,idx) = A(idx,idx) + Aloc; Array assembly I: growing arrays I = []; J = []; a = []];... t = idx(:,ones(length(idx),1)) ; I = [I;t(:)]; t = idx(:,ones(1,length(idx))); J = [J;t(:)]; a = [a; Aloc(:)]; Array assembly III see code fragment above Time [s] Timing for different assembly routines # Dofs Sparse assembly Array assembly, type 1 Array assembly, type 2 Array assembly, type 3 Fig Local computations First option: analytic evaluations 2.2 p. 104

105 We discuss bilinear form related to, triangular Lagrangian finite elements of degree p: K triangle: a K (u, v) := grad u grad v dx element stiffness matrix. K Use barycentric coordinate representations of local shape functions bi K α = κ α λ 1 1 λα 2 2 λα 3 α N 3 0, α =p grad bi K ( = κ α α 1 λ α λ α 2 2 λα 3 3 grad λ 1 + α 2 λ α 1 1 λα λ α 3 3 grad λ 2+ α N 3 0, α =p α 3 λ α ) 1 1 λα 2 2 λα grad λ 3. 3, κ α R, (2.2.5) (2.2.6) to evaluate K λ β 1 1 λβ 2 2 λβ 3 3 grad λ i grad λ j dx, i, j {1, 2, 3}, β k N. (2.2.7) 2.2 p. 105

106 If a 1, a 2, a 3 vertices of K (counterclockwise ordering): λ 1 (x) = 1 ( ( a 2)) ( ) x 1 PSfrag a 2 2 K a2 2 2 a3 replacements 2 a1 3, a2 1 λ 2 (x) = 1 ( ( )) ( ) a 3 x 1 a 3 2 K a2 3 2 a1 2 a1 1, a3 1 λ 3 (x) = 1 ( ( a 1)) ( x a 1 2 K a2 1 a2 2 ) a1 2. a1 1 ω 3 a 3 = ( a 3 1, a3 2 ω 1 ω 2 a 1 = ( a 1 1, a1 2) T a 2 = ( a 2 1, a2 2 ) T ) T grad λ 1 = 1 2 K ( ) a2 2 a3 2 a1 3 a2 1, grad λ 2 = 1 2 K ( ) a2 3 a1 2 a1 1 a3 1, grad λ 3 = 1 2 K ( a 1 2 a2 2 ) a 2 1 a1 1. ( K ) 3 grad λ i grad λ j dx i, j=1 = = 1 2 cot ω 3 + cot ω 2 cot ω 3 cot ω 2 cot ω 3 cot ω 3 + cot ω 1 cot ω 1. (2.2.8) cot ω 2 cot ω 1 cot ω 2 + cot ω p. 106

107 Exercise. Lemma (Integration of powers of barycentric coordinate functions). For any nondegenerate d-simplex K and α j N, j = 1,..., d + 1, λ α 1 1 λα d+1 d+1 dx = d! K α 1!α 2! α d+1! (α 1 + α α d+1 + d)! K α N d+1 0. (2.2.9) Proof for d = 2 Appendix.1 Remark. Alternative: symbolic computing (MAPLE, Mathematica) for local computations Numerical quadrature Second option (for local evaluations): Numerical quadrature f (x) dx 2 K P K K M l=1 ω K l f (π K l ), π K l K, ω K l R. (2.2.10) 2.2 p. 107

108 Terminology: ω K l weights, π K l quadrature nodes (2.2.10) = local quadrature rule Mandatory for computation of load vector ( f complicated/only available in procedural form) for computation of stiffness matrix, if σ = σ (x) does not permit analytic integration. Guideline: only quadrature rules with positive weights are numerically stable. Parametric definition of local quadrature rules on reference cell K : K P f ( x) d x ω l f ( π l ) l=1 How to gauge the quality of parametric local quadrature rules? f (x) dx P 2 K ωl K f (πl K ) K M l=1 with ω K l = ω l, π K l = K ( π l ). Quality of a parametric local quadrature rule on K largest space of polynomials on K integrated exactly by the corresponding quadrature rule on K. 2.2 p. 108

109 Parlance: Quadrature rule exact for P p ( K ) Example (Local quadrature rules on triangles). quadrature rule of order p + 1 degree of exactness p If K triangle K := convex { (00 ), ( 10 ), ( 01 ) }. Quadrature rules described by pairs ( ω 1, π 1 ),..., ( ω P, π P ), P N. Quadrature rule of order 2 (exact for P 1 ( K )) {( ( )) 1 0 6,, 0 ( 1 6, ( )) ( 0 1, 1 6, ( ))} 1 0. (2.2.11) Quadrature rule of order 3 (exact for P 2 ( K )) {( ( )) ( ( )) 1 1 6, /2 1 0, 0 6,, 1/2 ( 1 6, ( 1 /2 1/2 ))}. (2.2.12) One-point quadrature rule of order 2 (exact for P 1 ( K )) {( ( ))} 1 1 2, /3 1/3. (2.2.13) 2.2 p. 109

110 Quadrature rule of order 6 (exact for P 5 ( K )) { ( ( )) ( , / ( ) ) ( /21,, 1/ ,, 15/ ( ( ) ) ( /21, ,, 15/ ( 155 ( ) ) ( /21, ,, 15/ ( 6 15 / /21 ( 6 15 / /21 ( / /21 ) ), ) ), ) )} (2.2.14) Example (Local quadrature rules on quadrilaterals). Fig p. 110

111 If K quadrilateral K := convex { (00 ), ( 10 ), ( 01 ), ( 11 ) } (unit square). On K : tensor product construction: If {(ω 1, π 1 ),..., (ω P, π P )}, P N, quadrature rule on the interval ]0, 1[, exact for P p ]0, 1[, then { (ω1 2, ( π ) 1 π ) (ω1 ω 1 P, ( π ) 1 π ) P.. (ω 1 ω P, ( π ) P π ) (ω 2 1 P, ( π ) } P π ) P quadrature rule on K, exact for Q p ( K ). 2.2 p. 111

112 20 Gauss Legendre nodes in [ 1,1] Quadrature rules on ]0, 1[ ( basic numerics): classical Newton-Cotes formulas (equidistant quadrature nodes). Gauss-Legendre quadrature rules, exact for P 2P (]0, 1[) using only P nodes. Gauss-Lobatto quadrature rules: P nodes including {0, 1}, exact for P 2P 1 (]0, 1[). Number P of quadrature nodes t Fig Treatment of essential boundary conditions Remember Sect. 1.8: extension g g of Dirichlet data into yielded linear variational problem. Adaptation to finite element setting: 2.2 p. 112

113 V N = finite element space without constraints on. FIRST STEP: Interpolation/projection of boundary data FE-space V N W N := V N (FE trace space) Example: if V N = S 0 1 (M), then W N = set of piecewise linear, continuous functions on boundary mesh M. BUT, not necessarily g W N! Replace g by (interpolant, leaast squares fit, etc.) g N W N Example: if V N = S 0 1 (M) and g C0 ( ), then choose g N as p.w. linear interpolant. SECOND STEP: Trivial extension of g N g N V N Only nodal basis functions associated with node/edge/face contribute to g N! 2.2 p. 113

114 Example: if V N = S 0 1 (M), g N p.w. linear continuous on M g N = p N (M ) g N(p) b p N, where bp N = hat function for node p. u N V N,0 : a(u N + g N, v N ) = f (v N ) v N V N,0. (2.2.15) V N,0 := {v N V N : v N = 0 on } = span of interior basis functions. Equivalent algebraic perspective: partitioning of (big, w.r.t. V N ) linear system A µ = ϕ ( ) ( ) ( ) A A Ɣ µ ϕ =. (2.2.16) A Ɣ A ƔƔ µ Ɣ ϕ Ɣ Note: coefficients µ Ɣ known = relevant coefficients of g N Elimination A µ = ϕ A Ɣ µ Ɣ. (2.2.17) Remark Alternative: elimination on element level modified ϕ K ( ) ( ) Aii A A K = bi ϕi, ϕ A ib A K = à bb ϕ K = A ii, ϕk = ϕ i A bi µ Ɣ,K. b Then do assembly based on à K and ϕ K. p

115 2.2.8 Boundary approximation Sect approximate treatment of curved by parametric FE: Idea: Piecewise polynomial approximation of PSfrag replacements boundary (boundary fitting) ( locally considered as function over straight edge of an element) a 2 K E Ɣ n δ Example: Piecewise quadratic boundary approximation (Part of between a 1 and a 2 a 1 approximated by parabola) a 3 Mapping K curved element K : ( x) := x + 4δ λ 1 ( x)λ 2 ( x) n. (λ i barycentric coordinate functions on K, n normal to E Ɣ ) Note: Essential: diffeomorphism δ sufficiently small 2.2 p. 115

116 Transformation formula for gradients: for u : K R, diffeomorphism : K K (grad x ( u))( x) = (D ( x)) T (grad x u)( ( x)) x K. (2.2.18) Proof : chain rule: u (x) = d u( 1 u (x)) = ( 1 (x)) 1 j (x). x i x i x j=1 j x i ( T grad u(x) = D (x)) 1 grad x ( u)( 1 (x)) = D ( 1 (x)) T (grad u)( 1 (x)). Parametric construction: b K i = bi K, i = 1,..., Q Local shape functions on K Local shape functions on K Local computations use (2.2.18) & transformation formula (for multidimensional integrals): f ( (x)) dx = f ( x) det D ( x) d x for f : K R, K K 2.2 p. 116

117 K grad u grad vdx = = K K (grad u)( ( x)) (grad v)( ( x)) det D ( x) d x D T ( x) grad x ( u) D T ( x) grad x ( v) det D ( x) d x. K grad b K i grad b K j dx = K { D ( x) T D ( x)} 1 grad b K i grad b K j det D ( x) d x. Note: local shape functions b K i simple polynomials! For parabolic boundary fitting: D = I d + 4δ n grad(λ 1 λ 2 ) T R 2,2, det(d ) = 1 + 4δ n grad(λ 1 λ 2 ). Next: numerical quadrature ( Sect ) on K 2.2 p. 117

118 2.2.9 Treatment of hanging nodes Reminder: 2D: if edge of a cell is union of edges of other cells hanging node(s). a 1 K 1 F 1 ζ K 2 Discussion for H 1 ( )-conforming FE model mesh K F F 2 a 2 Fig. 37 Crucial: Global continuity of FE functions Special case: Linear Lagrangian finite elements on model mesh M, Fig. 37 v N S1 0 (M): v N (ζ ) = ζ a1 a 2 a 1 v N (a 2 ) + ζ a2 a 2 a 1 v N (a 1 ). Then, if v N K P 1 (K ) K M v N continuous across F. (= linear interpolation from a 1, a 2 ζ ζ slave node ) Assembly ( Sect ) (for V N = S1 0(M) on mesh with hanging nodes): 2.2 p. 118

119 Assumption: mesh data structure ( Sect ) of Ex (Coordinates, Elements) slave nodes = midpoints of (master) edges extra information on masters (a 1, a 2 in Fig. 37) of slave nodes (ζ in Fig. 37): slaves(:,1) ˆ= indices of slave nodes in coordinates slaves(:,2:3) ˆ= indices of master nodes ➊: Assembly treating slave nodes and regular nodes alike stiffness matrix à ➋: Add 1 2 slave row of A to related master rows : Post-processing of stiffness matrix for slave nodes N = size(a,1); S = speye(n) + 0.5*sparse([slave(:,2);slave(:,3)],... [slave(:,1);slave(:,1)],... ones(2*size(slave,1)),n,n); A = S*A*S ; A(slave(:,1),:) = []; A(:,slave(:,1)) = []; A change of basis! Lemma Deleting rows/columns slave nodes Note: More efficient: process slave nodes during local assembly p

120 Static condensation interior basis functions = global shape functions supported inside a cell (occur for S3 0 (M) on triangular mesh M in 2D) Sorting of global basis functions: coefficients for interior basis functions last Block structure of resulting linear system A µ = ϕ ( ) ( ) Aoo A A µ = oi µo A io A ii µ i ( ) ϕo = ϕ i = ϕ. (2.2.19) A ii coupling among interior basis functions A oi coupling between interior b.f. & basis functions on nodes/edges Note: A ii is block-diagonal with small blocks easy to invert [Elimination of µ i (Static condensation)] Schur complement system: ( A oo A oi A 1 ii A io ) µ o = ϕ o A oi A 1 ii ϕ i. 2.3 p. 120

121 2.3 Finite Difference Methods (FDM) Finite difference methods: Perspective: classical interpretation of boundary value problems Idea: replace derivatives by difference quotients using values at nodes of a grid = regular mesh From FEM to FD Model problem ( Sect. 1.2): 1D heat conduction, σ = σ (x), homogeneous Dirichlet b.c. ( d σ (x) d ) dx dx u = f in :=]0, 1[, u(0) = u(1) = 0. Assumption: σ C 0 (]0, 1[), f C 0 (]0, 1[]). Finite element Galerkin discretization Sect 2.1.3: p

122 M = {]ih, (i + 1)h[, i = 0,..., N,N N: equidistant mesh with meshwidth h := (N + 1) 1, finite element space V N = S 0 1,0 (M) (p.w. linear Lagrangian FE), S0 1,0 (M) H1 0 ( ). Local shape functions on cell K =]ih, (i + 1)h[, i = 0,..., N : Ex b1 K b2 K (x) = 1 x ih (x) = x ih h, h, d dx bk 1 (x) = 1 h, d dx bk 2 (x) = 1 h. Local computations based on numerical quadrature ( Sect ): element stiffness matrix: midpoint quadrature K f (x) dx h f (ξ i+1/2), ξ i+ 1/2 := (i + 1/2)h, element load vector: trapezoidal rule K f (x) dx 1 2 h( f (ξ i) + f (ξ i+1 )), ξ i := ih. Element stiffness matrix A K = ( hσ (ξ i+ 1/2) ( d dx bk k ) (ξi+ 1/2) ( d dx bk l ) (ξi+ 1/2)) 2 l,k=1 = 1 h σ i+1/2 ( 1 ) R 2, p. 122

123 Element load vector: ϕ K = 1 2 h ( ) 2 f (ξ i )b K j (ξ i) + f (ξ i+1 )b K j (ξ i+1)) = 1 j=1 2 h ( ) f (ξi ) f (ξ i+1 ) R 2. }{{} Tridiagonal linear system of equations A µ = ϕ, where µ i u(ξ i ): σ 12 + σ 32 σ σ 32 σ 32 + σ 52 σ 52 0 A = 1 0 σ 52 σ 52 + σ 72 σ 72 0 h σ N 3 σ 2 N 3 + σ 2 N 1 σ 2 N σ N 1 σ 2 N 1 + σ 2 N+ 1 2, (2.3.1) ϕ = h ( f 1 f 2 f N ) T R N, (2.3.2) (σ i+ 1/2 := σ (ξ i+ 1/2), f i := f (ξ i )). 2.3 p. 123

124 FDM approach: Approximation of derivative by symmetric difference quotient: d dx u u(ξ h) u(ξ 1 2 h) x=ξ h Applied twice at positions x = ξ i, x = ξ i+ 1/2, x = ξ i 1/2:. d d (σ (x) dx dx u) 1 x=ξ i h ( σ i+ 1/2 d dx u x=ξ i+ 1/2 Use this approximation at grid points ξ i, i = 1,..., N : 1 ) d σ i 1/2 dx u x=ξ i 1/2 1 h 2 ( σi+ 1/2(u(ξ i+1 ) u(ξ i )) σ i 1/2(u(ξ i ) u(ξ i 1 )) ). h 2 ( σi+ 1/2(u(ξ i+1 ) u(ξ i )) σ i 1/2(u(ξ i ) u(ξ i 1 )) ) = f i, i = 1,..., N. (2.3.3) Setting µ i := u(ξ i ) (2.3.3) = h 1 A µ = h 1 ϕ from (2.3.1), (2.3.2). 2.3 p. 124

125 Stencil notation for (row of) linear system (2.3.1), (2.3.2): h 1 [ σ i 1/2 σ i 1/2 + σ i+ 1/2 σ i+ 1/2 ξ i 1 ξ i ξ i+1 ] h,x=ξ i µ = h [ f i ] h. 1 ( ) σi 1/2µ i 1 + (σ i 1/2 + σ i+ 1/2)µ i σ i+ 1/2µ i+1 = h fi, i = 1,..., N. h Convention: µ 0 = µ N+1 = 0 Homogeneous Dirichlet BVP for Laplacian: u = f in :=]0, 1[ 2, u = 0 on. Finite element Galerkin discretization: M = triangular tensor-product grid (meshwidth h = (1 + N) 1, N N) V N = S1,0 0 (M), piecewise linear Lagrangian finite elments Sect Global shape functions: hat functions 2.3 p. 125

126 ents h a 3 K a 1 a 2 h Element stiffness matrix from (2.2.8): A K = ( numbering of local shape functions) Element load vector: use three-point quadrature formula (2.2.11) f (a 1 ) ϕ K = 1 6 h2 f (a 2 ). f (a 3 ) 2.3 p. 126

127 ➂ ➂ ➁ ➂ green: local vertex numbers K 5 Contributions to load vector component associated with node p: nts ➂ ➀ K 1 K 6 ➁ ➀ ➁ ➀ p ➂ ➂ K 3 K 4 ➁ ➁ From K 1 : ( ϕ K1 ) 2 From K 2 : ( ϕ K2 ) 3 From K 3 : ( ϕ K3 ) 3 From K 4 : ( ϕ K4 ) 1 From K 5 : ( ϕ K5 ) 1 From K 6 : ( ϕ K6 ) 2 K 2 ➀ ➀ ➁ ➀ ϕ p = h 2 f (p). 2.3 p. 127

128 ➂ ➂ ➁ ➂ ➂ ➀ ➁ ➁ ➂ ➀ ➂ 1 2 ➀ ➁ ➁ ➀ ➀ ➁ ➀ 2.3 p. 128

129 ➂ ➀ ➁ ➂ ➁ ➂ ➂ ➁ ➀ ➀ ➂ ➂ ➁ ➁ ➀ ➀ ➁ ➀ 2.3 p. 129

130 center node p grid position (ih, jh), 1 i, j N Indexing scheme for coefficient vector: µ i, j hat function at (ih, jh) (1 i, j N) Row of A µ = ϕ node p: 1 4µ i, j µ i 1, j µ i+1, j µ i, j 1 µ i, j+1 = h 2 f i, j. (µ i, j = 0 i = 0, N + 1 j = 0, N + 1) 5-point stencil for : h,p µ = h 2 f (p). (2.3.4) 2.3 p. 130

131 Finite difference approach to : approximation by symmetric difference quotients d 2 dx 2u x=(ξ,η) d 2 dy 2u x=(ξ,η) u(ξ h, η) 2u(ξ, η) + u(ξ + h, η) h 2, u(ξ, η h) 2u(ξ, η) + u(ξ, η + h) h 2. u x=(ξ,η) 1 h 2 ( 4u(ξ, η) u(ξ h, η) u(ξ + h, η) u(ξ, η h) u(ξ, η + h) ). Using this approximation at grid point p = (ih, jh): (modulo scaling by h 2 ) 5-point stencil (2.3.4) (Most) finite difference schemes finite element Galerkin schemes with numerical quadrature on structured meshes More on the stencil notation: 2.3 p. 131

132 α nw α w α n α c h αne α e 9-point stencil (on tensor product grid with meshwidth h) α nw α n α ne α w α c α e α sw α s α se h Terminology: α x = weight α c = center off-center α sw α s α se h Stencils offer a description of a finite dimensional linear operator without imposing an order on the vector components (Matrix notation requires ordering!) (Difference) stencils local linear operators on grid function space Bilinear form a(, ) on finite element space V N, see (2.1.1) [Choosing nodal FE basis functions] 2.3 p. 132

133 Local linear operators on grid function space ( stencils) [Ordering the basis] Stiffness matrix A Example: skew stencil α ne α w α c α e α sw 1/2h,1/2 3h h Stencils also meaningful on unstructured meshes PSfrag replacements p 2.3 p. 133

134 2.3.2 The discrete maximum principle Linear 2nd-order elliptic BVP, inhomogeneous Dirichlet problem, variational form u g + H0 1 ( ): σ grad u grad v dx = f v dx v H0 1 ( ). (2.3.5) ( g = extension of Dirichlet data g, Sect. 1.8) FE Galerkin discretization finite difference scheme (based on mesh M) Choice of nodal global shape functions: unkowns µ i, i = 1,..., N, approximate u(p) at certain geometric mesh locations (e.g. nodes, midpoints of edges, etc.) If g C 0 ( ) & f 0, we know (maximum principle Sect 1.4): True also for discrete solution u N? min g(x) min u(x) max u(x) max g(x). x x x x 2.3 p. 134

135 Definition (Discrete maximum principle). A local linear operator on grid function space satisfies the discrete maximum principle, if, for all its stencils, (i) all their off-center weights are non-positive, (ii) there is a strictly negative off-center weight, (iii) the sum of all weights is non-negative. Mere sign conditions, easily verified If A = (a i j ) R N,N corresponding to local linear operator satisfying discrete maximum principle: a i j 0 i j, 1 i, j N, N a i j 0 i = 1,..., N, j=1 i {1,..., N}: j {1,..., N}: a i j < 0 a ii > 0. (2.3.7) Example: homogeneous Dirichlet BVP for on R 2, piecewise linear Lagrangian finite elements ( Sect 2.1.4), triangular mesh M of R p. 135

136 (2.2.8) element stiffness matrix (for triangle with interior angles ω i, i = 1, 2, 3): A K = 1 cot ω 3 + cot ω 2 cot ω 3 cot ω 2 cot ω 3 cot ω 3 + cot ω 1 cot ω 1. (2.2.8) 2 cot ω 2 cot ω 1 cot ω 2 + cot ω 1 p i K K ω i j Entry (A) i j of stiffness matrix (w.r.t. S1 0 (M)) corresponding to nodes p i, p j, 1 i, j N := N (M): ω i j (A) i j = 1 2 sin ω i j + sin ω i j sin ω i j sin ω i j. p j Note: N j=1 (A) i j = 0 i = 1,..., N. (2.3.7) for A angle conditions: sum of angles facing interior edge π, angles facing boundary edges π/2. (for non-dirichlet boundary conditions) 2.3 p. 136

137 Example of triangular meshes satisfying angle condition: Delaunay triangulations ( see help for MATLAB delaunay command for more explanations) Definition (Structurally symmetric matrix). A matrix A = (a i j ) R N,N is structurally symmetric, if a i j 0 a ji 0 i, j {1,..., N}. Theorem If A R N,N is structurally symmetric, regular and satisfies (2.3.7), then A 1 0 componentwise. Proof. By permutation: A in block-diagonal form with connected blocks A 11 0 A = 0 A A J J. 2.3 p. 137

138 Then, focus on a single connected diagonal block (call it A = (a i j ) R M,M ): ( A invertible! ) For j {1,..., M}: µ R M : A µ = ɛ j := (δ i j ) M i=1 k {1,..., M}: µ k = min i=1,...,m µ i. Assumption: µ k < 0 0 a kk µ k + a k j µ j = a kk µ k j k,a kj 0 ( ): <, if µ j > µ k µ j = µ k, if a k j 0. j k,a kj 0 a k j µ j ( ) µ k M j=1 a k j 0. All other coefficients of µ connected with index k have to be equal to µ k. (A connected) µ = µ k 1 = (µ k,..., µ k ) T µ k A 1 }{{} 0 componentwise 0 componentwise. Contradicts A µ = ɛ j µ 0 componentwise. Revisit: Inhomogeneous Dirichlet problem (2.3.5) with f 0, σ 1, 2.3 FE Galerkin discretization by Lagrangian FEM, nodal global shape functions p. 138

139 Related linear system of equations (see Sect , (2.2.17)): A = ( ) A A Ɣ A Ɣ A ƔƔ µ Ɣ = values g j of Dirichlet data g in nodes. A µ = ϕ A Ɣ µ Ɣ. Note: N (A) i j = 0 A 1 = 0. j=1 µ Ɣ µ Ɣ γ 1 µ µ γ 1. Note: Matrix A regular (by Thm & Thm ) Assumption: stiffness matrix A (no boundary conditions!) satisfies (2.3.7) Apply Thm (<,>,, to be understoood in componentwise sense): µ Ɣ 0 A Ɣ 0 A Ɣ µ Ɣ 0 Thm µ = (A ) 1 A Ɣ µ Ɣ 0. µ Ɣ := µ Ɣ min{g j } 1 0 µ := µ 1 min{g j } 0 µ min{g j } µ Ɣ := max{g j } 1 µ Ɣ 0 µ := max{g j } 1 µ 0 µ max{g j } 1. p. 139

140 min{g j } (µ ) i max{g j } i = 1,..., N. Nodal values of u N satisfy (discrete) maximum principle! Finite difference convergence theory Perspective : classical interpretation of elliptic BVP, u C 2 ( ), f C 0 ( ) Functional framework : space C 0 (N ) = R N of grid functions f : N R, N = set of nodes Targetted norm : maximum norm on C 0 (N ) Tool : pointwise restriction operator R : C 0 ( ) C 0 (N ), (Ru)(p) := u(p). Reminder: The maxmimum (operator) norm of a matrix A R N,N is defined by A ξ A := ξ. sup ξ R N \{0} 2.3 p. 140

141 For A = (a i j ) R N,N : N A = max{ a i j, i = 1,..., N}. j=1 Linear system arising from a nodal FD/FE discretization of 2nd-order linear elliptic BVP L(u) = f (+boundary conditions, quadrature): A µ = ϕ := R f. (2.3.8) (Linear systems arising from Lagrangian FE (with nodal bases) require scaling to get (2.3.8)) Theorem (Lax theorem: consistency + stability convergence). If u C 2 ( ) is a classical solution of the BVP and A regular, then Ru µ A 1 ARu RL(u). Proof. By submultiplicativity of matrix norm: Ru µ A 1 ARu A µ + ϕ RL(u) = A 1 ARu RL(u). 2.3 p. 141

142 term ARu RL(u) is called consistency error Condition A 1 uniformly bounded in some discretization parameter: stability Theorem If A R N,N regular, A 1 0 componentwise, then ξ R N : A ξ 1 A 1 ξ. Proof. A 1 = sup A 1 η η =1 A 1 1 A 1 A ξ. Example (Convergence of simple FD). u = f in :=]0, 1[ 2, u = 0 on. Discretization: 5-point stencil on tensor product grid N = {(ih, jh), 1 i, j N, h := (N + 1) 1 Resulting A satisfies discrete maximum principle Sect , Def p. 142

143 Comparison function w(x) := 1 2 x 1(1 x 1 ) A(Rw) 1 pointwise. (by Thm ) A for any dimension N /meshwidth h (ie. uniformly )! Uniform stability of 5-point stencil for independently of meshwidth of tensor product grid Tackling consistency error: tool: Taylor expansion u(ξ±h, η) = u(x)± u (x) h u x 2 1 x 2 1 u(ξ, η±h) = u(x)± u (x) h u x 2 2 x 2 2 h 2 ± 1 3 u 6 x 3 1 h 2 ± 1 3 u 6 x 3 2 h h ξ±h ξ η±h η (ξ ± h t) 3 4 u x1 4 (t, η) dt, (η ± h t) 3 4 u x2 4 (ξ, t) dt. 1 h 2 ( 4u(ξ, η) u(ξ h, η) u(ξ + h, η) u(ξ, η h) u(ξ, η + h) ) = = 2 u x u x R(ξ, η, h), 2.3 p. 143

144 { with remainder R(ξ, η, h) u h2 4 u max, x 4 1 x2 4 } (ξ, η), h > 0. Consistency term for 5-point stencil for, tensor product grid, u C 4 ( ): { A(Ru) + R u 1 4 u 6 h2 4 } u max, N N. x 4 1 x2 4 Asymptotic convergence estimate: Ru µ Ch 2 with C = C(, u) > 0 independent of meshwidth h > p. 144

145 2.4 Finite Volume Methods (FVM) Principles of FVM Targeted problem: linear 2nd-order elliptic boundary value problem in 2D Sect. 1.1 div(σ grad u) = f in, u = 0 on. First ingredient: control volumes 2.4 p. 145

146 nts p j C j C k Ɣ ik. p k n ik Control volumes = C i p i (polygonal) cells of a mesh M = {C i } i covering computational domain. Associate cell C i nodal value µ i Meaning: µ i u(p i ), p i = center of C i Fig. 38 Second ingredient: numerical fluxes Two adjacent cells C k, C i with common edge Ɣ ik := C i C k. Numerical flux J ik = (µ i, µ k ) Ɣ j n ik ik ds 2.4 ( = numerical flux function, j = flux, see (1.2.1)) p. 146

147 By Gauss theorem conservation of energy (1.2.4): j n i ds = f dx J ik = C i f (p i ). C i C i k U i U i := {C j M: C i and C j share edge }, p i = node associated with control volume C i. System of equations ( M := M equations/unknowns µ i ): k U i (µ i, µ k ) = C i f (p i ) dx i = 1,..., M. (2.4.1) Note: homogeneous Dirichlet problem only interior control volumes in (2.4.1) Dual meshes widely used approach to construction of control volumes based on conventional FE triangulation M of. 2.4 (Here: triangular mesh of 2D polygon ) p. 147

148 First option: Voronoi dual mesh N (M) = {p 1,..., p M } = nodes of M C i := {x : x p i < x p j j i}. Voronoi dual mesh M := {C i } M i=1 ( MATLAB command voronoi) Fig. 39 Angle condition (Delaunay triangulation) Voronoi dual mesh 2.4 p. 148

149 Construction of Voronoi dual cells: edges perpendicular bisectors nodes circumcenters of triangles C i p i straightforward generalization to 3D PSfrag replacements Fig. 40 ω... Obtuse angle ω: circumcenter triangle Fig p. 149

150 Second option: Barycentric dual mesh Dual cells: edges union of lines connecting barycenters and midpoints of (primal) edges nodes barycenters of triangles No geometric obstructions Fig From FVM to FEM We consider: homogeneous Dirichlet problem for u = f in, u = 0 on. p

151 Finite volume method on Voronoi dual cells Fig. 39: M = triangular mesh of, only non-obtuse triangles angle condition Number of control volumes = number of interior nodes of M Numerical flux J ik := grad I 1 u n ik ds, Ɣ ik I 1 u S 0 1,0 (M) = M-piecewise linear interpolant of nodal values µ i (zero on ) Ɣ ik µ i µ ( k p k U i p k = i [grad(i 1 u) piecewise constant on triangles of M!] Ɣ ) ik µ i p i p k k U i k U i Ɣ ik p i p k µ k = C i f (p i ) i = 1,..., M. Entry of system matrix A (k U i, i = 1,..., N ): a ik = Ɣ ik p i p k, a ii = k U i Ɣ ik p i p k. 2.4 p. 151

152 Ɣ ik K p k Local perspective: Element stiffness matrix for finite volume scheme For triangle K p i Ɣik K := Ɣ ik K, Jik K (u) := grad I 1 u n ik ds. Ɣ K ik A K = J ik K (λ i) Ji K j (λ i) J ik K (λ i) Fig. 43 Jik K k) Ji K j (λ j) Jik K k) J jk K (λ k) J jk K (λ j) J K i j (λ i) J K jk (λ k) J K i j (λ j) J K jk (λ j) Note: Zero row sum 2.4 p. 152

153 With b i = hat function at node p i, E ik = edge connecting p i, p k, E i ik = [p i, 1/2(p i + p k )]: Then: integration by parts (1.6.3) on K, Gauss theorem on control volumes K. K grad I 1 u grad b i dx = = K = ω i ω j ω k grad I 1 u n K b i ds = E i ik Ei i j Ɣ K i j grad I 1 u n K ds grad I 1 u n i j ds p i Ɣ K ik E ik E i ik E k ik Ɣ K ik p k Ɣ K i j E i j Ɣ K k j E jk E ik E i j grad I 1 u n K b i ds grad I 1 u n ik ds p j Same matrix as for linear Lagrangian FE on M Sect Barycentric dual mesh? exercise 2.5 p. 153

154 2.5 Finite Element Convergence Theory Focus: Finite element Galerkin discretization of 2nd-order elliptic BVPs. Boundary value problems (heat conduction): Sect. 1.1 Variational formulation: Sect. 1.6 Some Sobolev spaces: Sect. 1.7 Abstract Galerkin discretization: Sect Lagrangian finite elements: Sects , 2.1.5, A priori error estimates Linear variational problem (1.7.1) & its Galerkin discretization ( Sect ): (V N V = discrete trial/test space, dim V N < ) u V : a(u, v) = f (v) v V u N V N : a(u N, v N ) = f (v N ) v N V N. 2.5 p. 154

155 A priori error estimates provide bounds for some norm of the discretization error u u N depending on 1. problem parameters 2. (the class of) problem data 3. discretization parameters For elliptic boundary value problems (1.2.6) + {(1.3.1),(1.3.2),(1.3.3)}, finite element scheme: Problem parameters = computational domain, boundary parts Ɣ D, Ɣ N, Ɣ R, heat conductivity σ, cooling coefficient q Problem data = Source term f, Dirichlet data g, Neumann data h Discretization parameters = mesh M ( meshwidth, mesh parameters), type of (H 1 ( )-conforming) finite element ( polynomial degree) a(, ) continuous (Def ) Existence & uniqueness of u, u N Assume a(, ) V -elliptic (Def ) f ( ) continuous (Def ) Quasi-optimality of u N u u N V C A γ inf v N V N \{0} u v N V. 2.5 p. 155

156 Switching V N V V N V, V N V N ( refinement ) u u N V u u N V How to achieve refinement of FE space? h-refinement: replace M (for V N ) M (for V N ) Example: regular refinement of triangular mesh in 2D 2D triangular mesh 2D triangular mesh 2D triangular mesh # Vertices : 45, # Elements : 64, # Edges : # Vertices : 153, # Elements : 256, # Edges : # Vertices : 561, # Elements : 1024, # Edges : p. 156

157 K T 3 T 4 T 1 T 2 Regular refinement of triangle K into four congruent triangles T 1, T 2, T 3, T 4 For h-refinement: asymptotic a priori error estimates in terms of meshwidth h M := max{diam K : K M}, diam K := max{ p q : p, q K } : u u N V C(problem data, problem parameters, mesh parameters, FE type) ɛ(h M ), with function ɛ : R + R +, ɛ(h) 0 for h 0. Definition (Order of convergence for h-refinement). In the case of h-refinement: convergence of order s, s R + : ɛ(h) = h s. p-refinement: replace V N := S 0 p (M), p N with V N := S0 p+1 (M) V N V N 2.5 p. 157

158 For p-refinement: asymptotic a priori error estimates in terms of polynomial degree p u u N V C(problem data, problem parameters, mesh M) ɛ(p), with function ɛ : N R +, ɛ(p) 0 for p. Combination of h-refinement and p-refinement? OF COURSE (hp-refinement, [7]) hp-refinement: Estimates in terms of which discretization parameter? General asymptotic a priori error estimate in terms of N := dim V N : u u N V C(problem data, problem parameters, mesh parameters, FE family) (N), with : N R +, (N) 0 for N. Definition (Convergence rate). (N) = O(N α ), α > 0 : algebraic convergence with rate α (N) = O(exp( γ N δ )), γ, δ > 0 : exponential convergence 2.5 p. 158

159 N 1 2 N 1 /2 exp( N 1 /3 exp( N 1 /5 ) exp( N 1 /3 /10) 0.05 ts Φ(N) PSfrag replacements Φ(N) N 1 2 N 1 /2 exp( N 1 /3 exp( N 1 /5 ) exp( N 1 /3 /10) N N Linear plot of qualitative convergence behavior: algebraic/exponential convergence rates Exponential convergence will always win (asymptotically) 2.5 p. 159

160 Φ(N) lacements N 2 N 1 N 1 /2 exp( N 1 /3 ) exp( N 1 /5 ) exp( N 1 /3 /10) N Log-linear plot of decrease of discretization error for algebraic/exponential convergence rates 2.5 p. 160

161 Φ(N) lacements N 2 N 1 N 1 /2 exp( N 1 /3 ) exp( N 1 /5 ) exp( N 1 /3 /10) N Log-log plot of decrease of discretization error for algebraic/exponential convergence rates 2.5 p. 161

162 A priori error estimates, how? Quasi-optimality of Galerkin solution a priori error estimates How to estimate best approximation error inf v N V N \{0} u v N V? Well, given solution u seek candidate function w N V N with Natural choice: u w N V inf v N V N \{0} u v N V. w N by interpolation/averaging of u BUT, exact solution u unknown? Known: u solves BVP + A priori informtion about problem parameters & problem data }{{} u will belong to a certain class of functions (e.g. subspace S V ) 2.5 p. 162

163 2.5.2 The Sobolev scales Recall: Interpolation error bound for equidistant polynomial interpolation ( f q)(a + k b a N ) = 0, k = 0,..., N := p 1 p P p([a, b]) max f (t) q(t) 1 a t b (p + 1)! (b a)p+1 max f (p+1) (t). a t b derivatives of function f : [a, b] R required: f C p+1 ([a, b]). Interpolation estimates hinge on smoothness of interpolands! How to measure smoothness of solution u of elliptic BVP? Definition (Higher order Sobolev norms). The m-th order Sobolev norm, m N 0, for u : R d R (sufficiently smooth) is defined by m u 2 m := D α u 2 dx, where D α α u u := x α 1 1. xα d d k=0 α N d, α =k 2.5 p. 163

164 Sobolev space (by completion of C ( ) ) H m ( ) := {v : R: v m < }. provide framework for variational formulation of elliptic BVP ( Sect. 1.7) Sobolev spaces provide norms m that measure smoothness of functions Sobolev scale:... H 3 ( ) H 2 ( ) H 1 ( ) L 2 ( ) Theorem (Sobolev embedding theorem). m > d 2 H m ( ) C 0 ( ) C = C( ) > 0: u C u m u H m ( ). Another Sobolev scale: 2.5 p. 164

165 Definition W m, -norm, m N 0, for u : R sufficiently smooth is defined by u m, := max 0 k m max sup D α u(x). α N d 0, α =m x Sobolev scale... W 3, ( ) W 2, ( ) W 1, ( ) W 0, ( ) = L ( ) Note: W m, ( ) is not a Hilbert space, but a Banach space ( functional analysis) The Bramble-Hilbert lemma Example: 1D, =]0, 1[, I 1 : C 0 ([0, 1]) P 1 (]0, 1[) linear interpolation in x = 0, 1 Taylor expansion for u C 2 ([0, 1]) u(0) = u(x) u (x)x u(1) = u(x) + u (x)(1 x) 1 2 x tu (t) dt, 1 x (1 t)u (t) dt. 2.5 p. 165

166 1 0 u I 1 u 2 dx = = = u(x) u(0)(1 x) u(1)x 2 dx (1 x) tu (t) dt x (1 t)u (t) dt dx 4 x x ( x x (1 x)2 t 2 dt u (t) 2 dt x2 (1 t) 2 dt 0 ( ((1 x)2 + x 2 ) u (t) 2 dt u (t) 2 dt ) dx x 1 x ) u (t) 2 dt dx ( 1 ) 1 /2 u I 1 u u (x) 2 dx u C ([0, 1]). 2.5 p. 166

167 Definition (Sobolev semi-norm). The m-th order Sobolev semi-norm, m N, for sufficiently smooth u : R is defined by u 2 m := α N d, α =m D α u 2 dx. Theorem (Bramble-Hilbert lemma). If R d bounded, p N, then C = C(, p) > 0: inf u q p+1 C u p+1 u H p+1 ( ). q P p ( ) Proof. by compactness arguments ( functional analysis). Polynomials of degree p can absorb all terms of order p in Sobolev norm Note: Thm = generalization of 2nd Poincaré-Friedrichs inequality Thm ! 2.5 p. 167

168 2.5.4 Transformation techniques cf. use of reference element for parametric construction of local shape function Sect Definition Given the local trial space S 0 p (K ) (see Sects , 2.1.7) (K = cell of a mesh) and nodes q 1,..., q Q, Q := dim S 0 p (K ), whose associated point evaluation functionals provide local degrees of freedom ( Def ) for S 0 p (K ), the nodal interpolation operator I p : C 0 (K ) S 0 p (K ) on K is defined by (I p u)(q i ) = u(q i ) i = 1,..., Q, u C 0 K. Example K = convex{a 1, a 2, a 3 } triangle, p = 1 linear interpolation: I 1 u K = u(a 1 )λ 1 + u(a 2 )λ 2 + u(a 3 )λ 3. (λ i, i = 1, 2, 3 = barycentric coordinate functions, Def ) Example K = triangle convex{a 1, a 2, a 3 }, p = 2 quadratic interpolation: 3 I 2 u K = λ i (1 2λ i ) u(a i ) + i=1 1 i< j 3 4λ i λ j u( 1 2 (ai + a j )). 2.5 p. 168

169 Goal: Asymptotic interpolation error estimates for I p, p N: C > 0: u I p u X C ɛ(h) u Y, for suitable Sobolev (semi-)norms X, Y. h : meshwidth of underlying triangulation ɛ(h) : convergence rate function, ɛ(h) 0 for h 0 C : constant depending on controllable mesh parameters. Main steps of transformation techniques: ➀ Localization: express u Ip u X through local contributions of cells K M. ➁ pullback K (u I pu) K, K M, to the reference cell K, K = K ( K ): relationship K (u I p u) X, K u Ip u X,K. ➂ Bramble-Hilbert argument on K : γ > 0 : û Î p û X, K γ û Y, K u, where Î p := K I p K. ➃ Determine the impact of the transformation on the Y -(semi-)norm. p

170 Chain of estimates: with constants C K,C K > 0, universal constant Ĉ > 0: u I p u 2 X ➀ = K M ➁ K M C K ➂ Ĉ K M C K (u I p u) K 2 X,K K (u I pu) 2 X, K K (u) 2 Y, K ➃ Ĉ K M C K C K u 2 Y,K. Need to know behavior of (local) norms under pullback! Concrete case: affine transformations K : K K, see Def K ( x) := F x + τ, F R d,d regular, τ R d 2.5 p. 170

171 Lemma If K : K K is an affine mapping x F x + τ, then, for all m N 0, ( ) m + d û m, K d m F m det(f) 1/2 u d m,k u H m (K ), ( ) m + d u m,k d m F 1 m det(f) 1/2 û d m, K u H m ( K ), with F denoting the matrix norm of F associated with the Euclidean vector norm. Proof (for m = 1). chain rule grad x (u K )( x) = (D K ( x)) T grad x u( K ( x)) x K. K Note: D K = F, det(d K ) = det F 0. grad x û 2 d x = (D K ( x)) T grad x u( K ( x)) 2 d x K = F T grad x u( K ( x)) 2 d x K ( ) = F T grad x u(x) 2 det F 1 dx K F 2 grad x u 2 det F 1 dx. K 2.5 ( ) = application of transformation formula for integrals. p. 171

172 Role reversal of K, K other estimate. F, det F : elusive quantities? NO determined by geometry of cell K Definition For cell K M define its diameter h K = diam K := sup{ x y, x, y K }, and the maximum radius of an inscribed ball h K r K := sup{r > 0 : x K : x y < r y K }. Ratio h K /r K = shape regularity measure ρ K of K. r K PSfrag replacements For triangle K : ρ K large K distorted Now: Focus on simplicial cells (triangles, tetrahedra) 2.5 p. 172

173 Lemma If the smallest angle of a triangle K is bounded from below by α > 0, then sin(α/2) 1 ρ K 2 sin(α/2) 1. α Proof. see figure l 1/2h K sin(α/2) l sin(α/2) = r K h K sin(α/2). PSfrag replacements h K r K. 2.5 p. 173

174 Lemma If K, K R d, d = 2, 3, are a generic non-degenerate simplices and K : K K, K ( x) := F ξ + τ, the associated bijective affine mapping, then ( hk h K ) d ρ 1 d K = h K rk d 1 h ḓ K F h K 2r K det(f) = K K hd K = 1 2 ρ K h K h K, h K rd 1 K = ( hk F 1 h K 2r K = 1 2 ρ K h K ) d ρ d 1 K, (2.5.1) h K h K. (2.5.2) Proof. det F = K K (ratio of volumes) (2.5.1) trivial z = center of the largest inscribed ball of K. Then, (2.5.2) from F = sup{ F x, x = 1} = 1 2 r 1 K sup{ ( x) (ẑ), x ẑ = 2r K } h K /2r K, because K ( x), K (ẑ) K. }{{} uniform shape regularity control of change of norms under K. 2.5 p. 174

175 Definition (Mesh parameters). With notations from Def : meshwidth h M := max{h K, K M}, shape regularity measure ρ M := max{ρ K, K M}, quasi-uniformity measure µ M := max{h K /h K, K, K M}. Standard choice of reference simplices: {( ) ( ) ( )} K := convex,, for d = 2, (2.5.3) K := convex 0, 0, 1, 0 for d = 3. (2.5.4) Corollary M = simplicial triangulation, K according to (2.5.3) and (2.5.4), respectively. Then the affine mappings K : K K, K ( ξ) := F K ξ + τ K, K M, satisfy ρ 1 d M µ d M h d M det(f K ) h d M, F K h M, F 1 K ρ M µ M h 1 M. 2.5 p. 175

176 2.5.5 Interpolation error estimates General formula for nodal Lagrangian interpolation operator I p : C 0 ( ) S 0 p (M): K M: I p u K = Q u(qi K ) bi K, (2.5.5) i=1 q K i b K i, i = 1,..., Q = local interpolation nodes in cell K M, see Sect ,, i = 1,..., Q = local shape functions: b K i (q K j ) = δ i j. I p local: u C 0 ( ), supp u K = I p u K = 0. Notation: Local nodal interpolation operators: I K p : C0 (K ) S 0 p (K ). Consider: parametric Lagrangian FE S 0 p (M), reference cell K, diffeomorphisms K : K K, K M: 2.5 Notation: Nodal interpolation operator on reference cell: Î p : C 0 ( K ) S 0 p ( K ). p. 176

177 Lemma (Pullback and nodal interpolation commute). C 0 (K ) I K p S 0 p (K ) C 0 ( K ) Îp S 0 p ( K ). Lemma K (IK p u) = Î p ( K u) u C0 (K ). Proof. By parametric definition of local shape functions on K, see Sect , Q K (IK p u) = u(qi K ) K bk i = i=1 Q ( K u)( qk i ) b i. i=1 Theorem On simplicial mesh M nodal interpolation operators I p : C 0 ( ) S 0 p (M) satisfy for 2 k p + 1, 0 m k C = C(k, m, ρ M ): u I p u m Ch k m M u k u H k ( ). 2.5 p. 177

178 Proof (For k = 2, m = 1): use transformation techniques and Bramble-Hilbert arguments Pick K M, u H 2 (K ) u C 0 (K ), see Thm M = triangular/tetrahedral mesh mappings K affine, K ( x) = F x + τ, see Def u I K p u 1,K ➊ ( 1 + d ➋ = ➌ = Lemma & Lemma applicable: ) d F 1 det F 1 /2 d K (u IK 1 ) 1, K ( ) 1 + d d F 1 det F 1 /2 û Î 1 û d 1, K ( ) 1 + d d F 1 det F 1 /2 d ) d F 1 det F 1 /2 ( ➍ 1 + d C I d ) inf (I d Î1 )(û q) 1, K q P 1 ( K ) inf û q 2, K q P 1 ( K ) ( ➎ 1 + d C I C BH d F 1 det F 1 /2 û d 2, K ( )( ) ➏ 1 + d 2 + d C I C BH d 3 F 1 F 2 u d d 2,K ➐ C(ρ K h 1 K ) h2 K u 2,K = C ρ K h K u 2,K. (2.5.6) 2.5 p. 178

179 ➊ Lemma for m = 1. ➋ Lemma ➌ Î 1 = projection onto P 1 ( K ) ( Def ). ➍ Sobolev embedding Thm : û C 1 û 2, K, and continuity Î1 û 2, K C 2 û C I := C 1 C 2. Lemma If dim V < and X, Y norms on V ( Def ), then C, C > 0: C v X v Y C v X v V. ➎ Bramble-Hilbert lemma Thm ➏ Lemma for m = 2. ➐ Lemma Finally, take squares and sum over all cells. Traingular cells with bad shape regularity (ρ K large ): very small/large angles 2.5 p. 179

180 However: Estimates of Thm not sharp anisotropic interpolation estimates [1] Inspect approximate interpolation constants C K := C K,2 := sup u H 2 (K ) sup u H 2 (K ) u I 1 u 1,K u 2,K, u I 1 u 0,K u 2,K, with triangle K := convex { (00 ), ( 10 ), ( px p y ) }. Sampling the space of triangles (modulo similarity) nts p y K 0 p x, p y 1. + Numerical computation of C K, C K,2 0 p x 1 (implementation by A. Inci) 2.5 p. 180

181 C K nts p x p y p x K p p y

182 log(c K ) 3 nts p x p y K ) p x Example (Linear interpolation on acute and obtuse triangles). p p y

183 triangle K := convex { (00 ) (, 10 ) ( 1, /2) } h, h > 0, u(x, y) = x(1 x), 0 < x < 1. u 2 2,K = h, u I 1u 2 1,K = h h h 1, u I 1 u 2 0,K = h C 2 K u I 1u 2 1,K u 2 2,K h 2, u I 1 u 2 0,K u 2 2,K = I 1 good approximation in H 1 (K )-norm I 1 bad approximation in H 1 (K )-norm Watch out for obtuse angles! 2.5 p. 183

184 C K, p x p y p. 184

185 However: inf u v N 1 u Ip u v N S 0 1 is possible! p(m) ts PSfrag replacements δ δ h h Elementary cell of bad mesh M bad Elementary cell of good mesh M good On bad mesh : sup u H 2 ( ) On good mesh : sup u H 2 ( ) u I 1 u 1 as h/δ, u 2 u I 1 u 1 uniformly bounded in h/δ. u 2 Yet, inf u v N 1 v N S1 0(M bad) inf u v N 1 u H 2 ( ). v N S1 0(M good) Elliptic regularity theory We consider: scalar second order elliptic BVP, see Sect. 1.1, (1.2.6) + boundary conditions p

186 Thm smoothness u H k ( ) required Question: What can we say about smoothness of solutions of 2nd-order elliptic BVPs? Example: 1D, =]0, 1[, coefficient σ 1, homogeneous Dirichlet boundary conditions u = f, u(0) = u(1) = 0. Obvious: f H k ( ) u H k+2 ( ) (a lifting theorem) Generalization to higher dimensions ( R d )? Theorem (Smooth elliptic lifting theorem). If is C -smooth, ie. possesses a local parameterization by C -functions, and σ C ( ), then, for any k N, u H 1 0 ( ) and div(σ grad u) H k ( ) u H 1 ( ), div(σ grad u) H k ( ), and grad u n = 0 on u H k+2 ( ). Remember: statements about Sobolev spaces = statements about their norms In Thm : u H k+2 ( ) means C = C(, σ ) > 0: u k+2 div(σ grad u) k p

187 ents What about non-smooth, discontinuous σ? ω 4 c 4 c 3 ω 3 ω 5 c 5 2D, polygon: corners c i angles ω i c 2 ω 2 u H0 1 ( ): u = f C ( ) How will corners affect smoothness of u? c 1 ω 1 ω 6 c p. 187

188 Local considerations at corner c i : nts φ = ω i r Introduce polar coordinates (r, φ) at c i Separation of variables ansatz u(r, φ) = u r (r)u φ (φ). φ c i φ = 0 u = 0 on u φ (0) = u φ (ω i ) = 0. In polar coordinates: u = 1 r u (r r r ) u r 2 φ 2 (u r (r)u φ (φ)) = 0 u(r, φ) = r λ (κ sin(λφ) + ν cos(λφ)), λ > 0, κ, ν R. [+ boundary condition u = 0] u(r, φ) = r λ ik sin(λ ik φ), λ ik = kπ ω i, k N. (2.5.7) Terminology: u from (2.5.7) is called Dirichlet corner singular function for at c i. p

189 Corner singular functions locally (in a neighborhood of c i ) satisfy the homogeneous PDE and boundary conditions. Singular functions for k = 1, 2, 3 at a corner with ω = 3π/ p. 189